THE  UNIVERSITY  OF  CHICAGO 
MATHEMATICAL  SERIES 

Eliakim  Hastings  Moore 

General  Editor 


THE  SCHOOL  OF  EDUCATION 
TEXTS  AND  MANUALS 

George  William  Myers 
Editor 


THIRD-YEAR  MATHEMATICS 
for    SECONDARY    SCHOOLS 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


THE  BAKER  &  TAYLOR  COMPANY 

NEW  YORK 

THE  CAMBRIDGE  UNIVERSITY  PRESS 

LONDON 

THE  MARUZEN-KABUSHIKI-KAISHA 

TOKYO,   OSAKA,    KYOTO,   FHKUOKA,   SENDAI 

THE  MISSION  BOOK  COMPANY 

SHANGHAI 


GOTTFRIED  WILHELM  LEIBNITZ 


GOTTFRIED  WILHELM  LEIBNITZ  was  born  at  Leip- 
zig in  1646  and  died  at  Hanover  in  1716.  His  was  the 
genius  that  was  both  precocious  and  universal.  Even 
from  early  childhood  he  overcame  the  most  trying  obstacles.  He 
proved  himself  of  extraordinary  qualities  before  he  was  twelve. 
Though  he  had  mastered  the  ordinary  texts  in  mathematics, 
philosophy,  theology,  and  law  before  he  was  twenty,  not  until 
he  reached  the  age  of  twenty-six,  when  he  was  sent  on  a  political 
errand  to  Paris,  where  he  made  the  acquaintance  of  Huygens, 
did  he  become  strongly  interested  in  mathematics.  He  achieved 
eminence  of  the  highest  order  in  mathematics,  philosophy,  the- 
ology, law,  and  languages.  For  forty  years  following  1676  he 
held  the  post  of  librarian  of  the  house  of  Brunswick  and  Hanover, 
earning  the  highest  honors  and  distinctions  in  the  services  of  his 
house  only  to  be  cast  aside  in  old  age  by  the  existing  head  of  the 
Brunswick  family  when  he  became  George  I  of  England. 

Leibnitz'  political  and  religious  papers  touching  affairs  of  the 
dynasty  from  1673-1713  constitute  an  indispensable  contribu- 
tion to  the  history  of  his  time. 

His  place  in  the  history  of  philosophy  is  even  larger  than  it 
is  in  mathematics.  The  Leibnitzian  system  of  philosophy  con- 
stitutes a  most  important  epoch  in  the  history  of  philosophical 
doctrines.  Leibnitz'  life  denies  the  prevailing  idea  that  mathe- 
matical genius  is  necessarily  narrow  and  specialized.  It  also 
furnishes  a  conspicuous  example  of  a  most  important  contribu- 
tor to  the  advance  of-  mathematics  who  was  not  by  profession, 
at  any  time  during  his  life,  a  teacher  of  the  science. 

His  chief  services  to  mathematical  science  consist  of  his  inde- 
pendent invention  of  the  language,  if  not  the  substance,  of  the 
differential  calculus,  his  work  on  osculating  curves,  his  funda- 
mental work  on  the  theory  of  envelopes,  his  explanation  of  the 
method  of  expansion  of  functions  by  indeterminate  coefficients, 
and  his  recognition  of  the  theory  of  determinants  and  his  devel- 
opmental work  on  the  theory. 

His  work  displays  great  skill  in  analysis,  but  like  the  work  of 
most  geniuses  it  is  unfinished  and  characterized  by  frequent 
errors.  But  he  blazed  out  many  new  routes  through  the  mathe- 
matical regions  which  men  of  lesser  genius  were  aided  later  in 
converting  into  comfortable  highways  and  thoroughfares. 

The  later  years  of  his  life  were  embittered  by  a  contest  with 
the  overzealous  friends  of  Newton  over  the  question  of  priority 
of  invention  of  the  calculus  as  between  him  and  Newton.  Sub- 
sequent times  have  seemed  to  settle  the  controversy  on  the  basis 
that  Leibnitz  and  Newton  were  independent  inventors  of  the 
calculus,  and  that  most  certainly  the  modern  notation  of  the 
calculus  is  due  to  Leibnitz.  For  more  detail  about  this  famous 
controversy  any  history  of  mathematics  may  be  consulted. 

In  character  Leibnitz  was  quick-tempered,  intolerant,  selfish, 
and  inordinately  conceited.  But  the  products  of  his  genius  will 
ever  adorn  and  enrich  the  pages  of  mathematical  and  philo- 
sophical history. 

[See  Ball's  or  Cajori's  or  Tropfke's  History.] 


GOTTFRIED  WILHELM  LEIBNITZ 


Third-Year  Mathematics 
for  Secondary   Schools 

With  Logarithmic  and  Trigonometric 
Tables  and  Mathematical  Formulas 


BY 

ERNST  R.  BRESLICH 

Head  of  the  Department  of  Mathematics  in  the  University 
High  School,  The  University  of  Chicago 


THE  UNIVERSITY  OF  CHICAGO  PRESS 
CHICAGO,  ILLINOIS 


Copyright  1917  By 
The  University  of  Chicago 


All  Rights  Reserved 


Published  September  191 7 
Second  Impression  January  1920 


u  -*jl?  *b  "^° 


•  •  • 


Composed  and  Printed  By 

The  University  of  Chicago  Press 

Chicago.  Illinois.  U.S.A. 


EDITOR'S  PREFACE 

This  third  unit  of  Mr.  Breslich's  course  in  general 
mathematics  for  high  schools  aims  primarily  to  carry- 
forward the  spirit  and  the  method  of  the  two  former 
volumes.  By  using  chapter  xv  at  the  beginning  of  the 
year  as  a  syllabus  for  reviewing  the  ground  covered  in  the 
previous  work  in  geometry,  this  volume  may  be  readily 
taken  up  by  classes  whose  previous  work  has  been  in 
standard  courses  of  algebra  and  geometry. 

To  aid  pupils  who  do  not  go  beyond  the  high  school, 
and  whose  first  two  years  of  training  have  been  in  corre- 
lated mathematics,  to  become  sufficiently  familiar  with 
the  standard  special  mathematical  methods  and  principles 
of  algebra,  geometry,  and  trigonometry,  as  such,  and  to 
command  the  existing  literature  of  these  branches,  the 
type  of  correlation  here  used  is  what  may  be  termed 
topical.  A  certain  type  of  subject-matter  is  allowed 
emphasis  for  a  sufficient  time  to  enable  pupils  to  master 
the  appropriate  type  of  methodology.  This  will  seem, to 
a  superficial  critic  a  departure  from  the  close  type  of 
correlation  of  the  two  preceding  texts.  This  variation  is, 
however,  intentional,  to  the  end  that  algebra,  geometry, 
and  trigonometry,  as  such,  may  be  firmly  grasped  by  the 
pupil  while  he  is  yet  in  high  school.  The  correct  descrip- 
tion of  the  prevailing  procedure  here  is:  Isolation  in 
details,  but  correlation  in  major  matters.  This  assures 
any  real  benefits  of  an  isolated  type  of  treatment  without 
losing  the  more  important  values  of  correlation.  Mathe- 
matical training  must  foster  both  concentration  and 
generalship.  Classroom  experience  has  verified  the  pro- 
priety of  this  type  of  correlation  for  third-year  classes. 

vii 

420821 


viii  EDITOR'S  PREFACE 

The  author  would  request  open-minded  teachers  to 
give  the  form  of  reconstructed  mathematics  herewith 
presented  a  fair  classroom  test.  He  will  gladly  accept 
the  issue  of  such  a  test.  Superintendents  and  principals 
should  feel  that  here  is  something  of  the  sort  their  spokes- 
men have  been  urging,  and  see  to  it  that  the  text  be 
given  a  fair  test.  Better  things  can  hardly  be  obtained 
except  through  the  testing  of  different  methods.  Of  the 
methods  deserving  of  a  classroom  test,  those  that  have 
proved  successful  in  particular  instances  are  most  worth 
while.  The  material  of  this  volume  belongs  to  this  class. 
May  its  friends  become  as  numerous  as  are  those  of  its 
companion  volumes! 

G.  W.  Myers 

Chicago,  III. 

August,  1917 


AUTHOR'S  PREFACE 

This  book  is  the  third  of  the  series  of  textbooks  on 
secondary  mathematics.  It  is  designed  primarily  as  a 
third  unit  of  a  year's  work  to  follow  the  first  two  unit- 
courses  worked  out  by  the  author  in  First-Year  Mathe- 
matics and  Second-Year  Mathematics.  It  completes  the 
study  of  high-school  algebra,  trigonometry,  and  solid 
geometry. 

In  accordance  with  the  general  plan  of  the  series,  the 
book  aims  to  teach  in  combination  mathematical  topics 
which  are  naturally  closely  related  to  each  other  even 
though  drawn  from  separate  mathematical  subjects. 

Such  an  arrangement  has  the  advantage  of  developing 
the  subject  of  secondary-school  mathematics  in  a  sequence 
which  is  both  psychological  and  logical.  Indeed,  the 
student's  understanding  of  the  meaning  and  the  utility 
of  the  subject  is  deepened  to  such  an  extent  that  he  is 
better  able  to  appreciate  the  scientific  character  of  mathe- 
matics than  when  he  is  studying  the  separate  subjects. 
The  result  is  that  he  is  more  disposed  to  continue  the 
study  of  mathematics. 

Through  proper-  correlation  the  whole  third-year  work 
can  be  better  motivated  and  becomes  more  concrete,  each 
subject  gaining  from  the  study  of  others.  For  example, 
the  student  appreciates  the  need  of  studying  the  theory 
of  logarithms  because  of  their  usefulness  as  a  tool  for 
solving  problems  in  trigonometry.  He  further  sees  that 
he  must  master  the  theory  of  exponents  in  order  that  he 
may  understand  the  fundamental  principles  of  the  theory 
of  logarithms. 

ix 


X  THIRD-YEAR  MATHEMATICS 

In  the  study  of  simultaneous  linear  and  quadratic 
equations,  in  connection  with  intersecting  straight  lines 
and  curves,  the  abstract  processes  of  solving  the  various 
types  of  systems  of  equations  are  represented  concretely 
and  are  easily  understood  and  remembered. 

Through  the  graphical  representation  of  equations  of 
the  form  y  =f(x),  where  f(x)  is  either  a  polynomial  or  a  trig- 
onometric function,  he  learns  to  appreciate  the  funda- 
mental concept  of  functional  correspondence,  which  is  of 
greatest  value  to  him  as  a  natural  introduction  to  analyti- 
cal geometry. 

Thus  correlation  removes  the  disadvantages  of  the 
topical  plan  without  losing  its  advantages.  It  arouses 
and  holds  the  student's  interest.  Many  students  who 
find  third-year  algebra  as  a  separate  subject  too  abstract 
and  uninteresting  take  great  delight  in  a  third-year 
course  in  which  algebra  and  trigonometry  are  correlated, 
because  they  enjoy  the  study  of  trigonometry  and  its 
applications. 

Special  attention  is  directed  to  the  following  features 
of  this  course : 

1.  Reviews  are  carried  on  at  frequent  intervals 
throughout  the  course.  Hence  the  introductory  review 
found  in  most  third-year  texts  has  been  omitted.  Since 
the  course  begins  with  new  work,  the  student  has  at  once 
the  exhilaration  of  taking  a  step  in  advance,  while  at  the 
same  time  he  is  reviewing  in  a  way  that  gives  him  a  new 
and  higher  view  of  the  subject-matter.  His  time  and 
effort  are  economized  through  the  avoidance  of  useless 
repetitions. 

2.  At  the  end  of  each  chapter  the  principal  facts  are 
summarized. 

3.  The  last  chapter  of  the  book  is  a  syllabus  of  all  the 
theorems  of  plane  and  solid  geometry  which  were  studied 


AUTHOR'S  PREFACE  xi 

in  the  preceding  courses.  This  may  be  used  for  reference 
purposes  while  the  student  is  taking  the  course.  It  will 
also  be  found  effective  as  a  basis  for  a  final  review  of  plane 
and  solid  geometry  as  such.  The  list,  at  the  end  of  the 
book,  which  gives  all  important  mathematical  formulas 
so  far  studied  serves  the  same  purpose. 

4.  The  material  has  been  carefully  selected  with  a 
view  to  stimulating  independent  thinking  and  to  prepar- 
ing the  student  for  collegiate  mathematics.  The  number 
of  supplementary  exercises  is  sufficiently  large  to  furnish 
the  drill  needed  to  enable  the  student  to  meet  the  require- 
ments of  college  entrance  examinations.  For  this  purpose 
many  problems  taken  from  entrance  examinations  of 
various  colleges  have  been  included. 

5.  The  historical  notes  distributed  throughout  the 
book  add  to  the  interest  of  the  student  in  genetic  phases 
of  the  work.  The  portraits  appearing  as  inserts  have  been 
taken  from  the  Philosophical  Portrait  Series,  published  by 
the  Open  Court  Publishing  Company,  Chicago. 

The  author  takes  great  pleasure  in  acknowledging  his 
indebtedness  to  his  colleagues  in  the  department  of 
mathematics  for  many  valuable  constructive  criticisms. 
He  is  indebted  also  to  Principal  F.  W.  Johnson,  of  the 
University  High  School,  and  to  Professor  Charles  H. 
Judd,  Director  of  the  School  of  Education,  for  sympathetic 
appreciation  and  encouragement  during  the  preparation 
of  the  manuscript. 

Ernst  R.  Breslich 


CONTENTS 

PAGE 

Study  Helps  for  Students xvii 

CHAPTER 

I.  Functions.    Equations  in  One  Unknown     .     .  1 

Linear  Function 4 

Direct  Variation 6 

Quadratic  Function 8 

Graphical  Solution  of  Equations  of  Degree  Higher 

than  the  Second 10 

Synthetic  Division.  Remainder  Theorem  ...  12 
Equations  of  Degree  Higher  than  the  Second  Solved 

by  Factoring  .     . N 16 

The  Function- 19 

x 

II.  Trigonometric  Functions 25 

Angles  in  General 25 

Values  of  the  Trigonometric  Functions  Found  by 

Means  of  a  Drawing 29 

Changes  of  the  Trigonometric  Functions      ...  31 

Graphs  of  the  Trigonometric  Functions  ....  37 

Trigonometric  Functions  of  Negative  Angles     .      .  44 

Trigonometric   Functions   of  \^a)  m  Terms  of 

the  Functions  of  a 46 

Trigonometric  Functions  of  (n  •  o±a)  m  Terms  of 

the  Functions  of  a 49 

III.  Linear  Equations 53 

Linear  Equations  in  One  Unknown 53 

Linear  Equations  in  Two  Unknowns  ....  61 
Solution    of  a  System  of   Linear   Equations    by 

Determinants 62 

Linear  Equations  with  Three  or  More  Unknowns  .  68 

Solution  by  Determinants  ....,.«,  70 
xiti 


xiv  CONTENTS 

CHAPTEB  PAGE 

•.-  IV.  Quadratic  Equations  in  One  Unknown  ...  76 

Methods  of  Solving  Quadratic  Equations     ...  76 

Square  Root  of  Polynomials 79 

Fractional  Equations    .     .     . 81 

Equations  of  Quadratic  Form 84 

Trigonometric  Equations 84 

Nature  of  the  Roots  of  a  Quadratic  Equation   .     .  86 
Relation  between  the  Roots  and  the  Coefficients 

of  a  Quadratic 90 

Factoring 91 

V.  Factoring.    Fractions 94 

The  Difference  of  Two  Squares          94 

The  Sum  or  Difference  of  Like  Powers    ....  95 

Trinomials 96 

Polynomials 96 

Fractions 98 

Complex  Fractions 99 

VI.  Exponents.    Radicals.    Irrational  Equations  104 
The  Fundamental  Laws  of  Positive  Integral  Ex- 
ponents        104 

Zero-Exponents.    Fractional  and  Negative  Expo- 
nents      108 

Radicals 114 

Reduction  of  Radicals 115 

Addition  and  Subtraction  of  Radicals       .     .     .     .  117 

Multiplication  of  Radicals .  118 

Division  of  Radicals           118 

-    Rationalizing  the  Denominator 120 

Square  Root  of  a  Radical  Expression       .     .     .     .  122 

Irrational  Equations 123 

Trigonometric  Equations 126 

VII.  Logarithms.    Slide  Rule 128 

Labor-Saving  Devices 128 

Precision  of  Measurement 128 

Logarithms  ............  131 


CONTENTS  XV 

CHAPTER  PAGE 

Common  Logarithms 134 

The  Table  of  Logarithms 136 

Properties  of  Logarithms 142 

-^Exponential  Equations 146 

The  Slide  Rule 147 

VIII.  Logarithms  of  the  Trigonometric  Functions. 

Solution  of  Triangles .     .  153 

Use  of  the  Table  of  Logarithmic  Functions       .     .  153 
Use  of  Logarithms  in  the  Solution  of  Right  Tri- 
angles   156 

Relations  between  the  Sides  and  Angles  of  Oblique 

Triangles 160 

Solution  of  Oblique  Triangles 168 

Area  of  an  Oblique  Triangle 176 

IX.  Relations    between    Functions    of    Several 

Angles 183 

Addition  and  Subtraction  Theorems       ....  183 

Functions  of  Double  an  Angle 189 

Functions  of  Half  an  Angle 191 

Trigonometric  Equations 193 

X.  Binomial  Theorem.    Arithmetical  and  Geomet- 
rical Progressions 196 

Binomial  Theorem 196 

Arithmetical  Progression* 202 

Geometrical  Progression 207 

Infinite  Geometrical  Series 212 

-   XL  Systems  of  Equations  in  Two  Unknowns  In- 
volving Quadratics 218 

Graphs  of  Quadratic  Equations  in  Two  Unknowns  218 

Solution  of  Simultaneous  Quadratics       ....  225 
Solution  of  Equations  of  Degree  Higher  than  the 

Second 231 

Solution  of  Irrational  and  Fractional  Equations    .  233 


xvi  CONTENTS 

CHAPTER  PAGE 

XII.  Areas  of  Surfaces 239 

Polyedrons.     Cylinders.    Cones 239 

Sections  Made  by  a  Plane 245 

Areas 255 

Surfaces  of  Revolution 264 

Area  of  the  Surface  of  a  Sphere 267 

XIII.  Volumes 272 

Volume  of  a  Rectangular  Parallelopiped       .     .     .  272 

Comparison  of  Volumes 275 

Volume  of  a  Prism 276 

Volume  of  a  Cylinder 281 

Volume  of  a  Pyramid 284 

Volume  of  a  Frustum  of  a  Pyramid 290 

Volume  of  a  Circular  Cone 291 

Volume  of  a  Frustum  of  a  Cone 292 

Volume  of  a  Sphere 293 

Volume  of  a  Spherical  Segment 295 

xiv.  polyedral  angles.    tetraedrons.    spherical 

Polygons 302 

Polyedral  Angles 302 

Tetraedrons 308 

Spherical  Angles 312 

Polar  Spherical  Triangles 314 

Symmetry  and  Congruence 317 

Area  of  a  Spherical  Triangle 323 

XV.  Assumptions  and  Theorems  of  Geometry  Given 

in  the  Courses  of  the  First  and  Second  Years  33 1 

Logarithms  of  Numbers 353 

Table  of  Powers  and  Roots 355 

Table  of  Sines,  Cosines,   and  Tangents  of  Angles 

from  l°-90° 356 

Formulas 357 

Reductions 362 

Index 365 


STUDY  HELPS  FOR  STUDENTS1 

The  habits  of  study  formed  in  school  are  of  greater  impor- 
tance than  the  subjects  mastered.  The  following  suggestions, 
if  carefully  followed,  will  help  you  make  your  mind  an  efficient 
tool.  Your  daily  aim  should  be  to  learn  your  lesson  in  less 
time,  or  to  learn  it  better  in  the  same  time. 

1.  Make  out  a  definite  daily  program,  arranging  for  a  definite 
time  for  the  study  of  mathematics.  You  will  thus  form  the 
habit  of  concentrating  your  thoughts  on  the  subject  at  that 
time. 

2.  Provide  yourself  with  the  material  the  lesson  requires; 
have  on  hand  textbook,  notebook,  ruler,  compass,  special 
paper  needed,  etc.  When  writing,  be  sure  to  have  the 
light  from  the  left  side. 

3.  Understand  the  lesson  assignment.  Learn  to  take  notes 
on  the  suggestions  given  by  the  teacher  when  the  lesson  is 
assigned.  Take  down  accurately  the  assignment  and  any 
references  given.  Pick  out  the  important  topics  of  the 
lesson  before  beginning  your  study. 

4.  Learn  to  use  your  textbook,  as  it  will  help  you  to  use  other 
books.  Therefore  understand  the  purpose  of  such  devices 
as  index,  footnotes,  etc.,  and  use  them  freely. 

5.  Do  not  lose  time  getting  ready  for  study.  Sit  down  and 
begin  to  work  at  once.  Concentrate  on  your  work,  i.e.,  put 
your  mind  on  it  and  let  nothing  disturb  you.  Have  the 
will  to  learn. 

1  These  study  helps  are  taken  from  Study  Helps  for  Students 
in  the  University  High  School.  They  have  been  found  to  be  very 
valuable  to  students  in  learning  how  to  study  and  to  teachers  in 
training  students  how  to  study  effectively. 

xvii 


xvin  STUDY  HELPS  FOR  STUDENTS 

6.  As  a  rule  it  is  best  to  go  over  the  lesson  quickly,  then  to  go 
over  it  again  carefully;  e.g.,  before  beginning  to  solve  a 
problem  read  it  through  and  be  sure  you  understand  what 
is  given  and  what  is  to  be  proved.  Keep  these  two  things 
clearly  in  mind  while  you  are  working  on  the  problem. 

7.  Do  individual  study.  Learn  to  form  your  own  judgments, 
to  work  your  own  problems.  Individual  study  is  honest 
study. 

8.  Try  to  put  the  facts  you  are  learning  into  practical  use  if 
possible.  Apply  them  to  present-day  conditions.  Illus- 
trate them  in  terms  familiar  to  you. 

9.  Take  an  interest  in  the  subject.  Read  the  corresponding 
literature  in  your  school  library.  Talk  to  your  parents 
about  your  school  work.  Discuss  with  them  points  that 
interest  you. 

10.  Review  your  lessons  frequently.     If  there  were  points  you 
did  not  understand,  the  review  will  help  you  to  master  them. 

11.  Prepare  each  lesson  every  day.    The  habit  of  meeting  each 
requirement  punctually  is  of  extreme  importance. 


CHAPTER  I 

FUNCTIONS.    EQUATIONS  IN  ONE  UNKNOWN 

Function.    Variable.     Constant 

1.  The  formula  i  =  prt  may  be  used  to  compute  the 
interest,  i,  of  a  principal,  p,  at  the  rate,  r,  for  t  years.  If 
the  rate  and  principal  remain  the  same,  the  interest,  i,  de- 
pends upon  the  time,  t,  in  the  sense  that  if  one  is  changed 
the  other  changes  correspondingly.  The  time  and  inter- 
est are  variables,  the  principal  and  rate  constants,  and  the 
interest  is  said  to  be  a  function  of  the  time. 

Dependence  of  one  magnitude  upon  another  is  met 
frequently.  For  example,  the  premium  of  a  life-insurance 
policy  depends  upon  the  age  of  the  applicant,  the  distance 
passed  over  by  a  moving  body  depends  upon  the  time, 
the  length  of  a  circle  depends  upon  the  radius. 

Sometimes  this  dependence  is  expressed  in  the  form 
of  an  equation.  Thus,  the  length  of  a  circle  is  given  by 
the  equation  c  =  2wr.  Because  of  this  equation,  to  every 
value  of  c  there  corresponds  a  definite  value  of  r.  This  is 
often  expressed  by  saying  that  c  is  a  function  of  r.  The 
symbols  c  and  r  are  variables,  tt  is  a  constant. 

2.  Constant.  A  symbol  which  represents  the  same 
number  throughout  a  discussion,  or  in  a  problem,  is  a 
constant. 

EXERCISE 

In  the  equations,  A  =  7rr2,  A  =  ^bh,  d=rt,  s  =  jgt2,  v  =  ^7rr3*, 
one  letter  depends  upon  one  or  more  other  letters  for  its  value. 
Which  symbols  in  these  equations  are  constants  ? 

*  When  such  forms  as  r2,  \bh,  rt,  IQt2,  2a; +3,  and  vV-25  first 
came  into  mathematics,  they  were  regarded  merely  as  shortened 

1 


2  TH'Itt  L'-YE  AK  MATHEMATICS 

3.  Variable.  A  variable  is  a  symbol  representing 
different  numbers  in  a  problem. 

Name  the  variables  in  the  foregoing  equations. 

4.  Function.  If  two  variables  x  and  y  are  so  related 
that  to  every  value  of  x  there  corresponds  a  definite  value 
of  y,  then  y  is  said  to  be  a  function  of  x. 

EXERCISES 

In  the  following  relations  show  that  one  symbol  is  a  func- 
tion of  one  or  more  others. 


1.    s  =  ^(a+b+c) 

4.  v=hh 

o 

2.  A=  cr 

5.  7  =  ^ 

o 

3.  A-tvi 

6.  C  =  ~(F-32) 

7.  Name  the  constants  and  the  variables  in  exercises  1-6. 

5.  Functional  notation.  Sometimes  we  are  interested 
in  the  relation*  between  two  variables  rather  than  in  the 
variables  themselves.  For  example,  in  uniform  motion 
the  distance  is  equal  to  the  rate  multiplied  by  the  time, 
but  with  falling  bodies  the  distance  is  approximately  16 
multiplied  by  the  square  of  the  time.  The  first  of  these 
laws  is  expressed  in  the  form  d  =  rt,  the  second  in  the  form 
d=  16Z2.     In  these  two  cases  d  is  not  the  same  function  of  t. 


statements  of  rules  of  reckoning,  just  as  in  percentage  we  regarded 
j)  =  b-r  as  a  short  way  of  stating:  percentage  equals  base  times  rate. 

Later  these  forms  came  to  be  regarded  as  either  (1)  rules  of 
reckoning,  or  (2)  the  results  of  the  reckoning.  As  results,  they  were 
regarded  as  numbers,  and  could  be  added,  subtracted,  multiplied, 
and  divided.     This  gave  rise  at  once  to  modern  algebra. 

*  Relation  means  any  interdependence,  and  not  necessarily  ratio. 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN       3 

The  equation  f=2n  expresses  the  relation  between  the 
number  of  miles  and  the  railroad  fare  at  2  cents  a  mile. 
The  relation  between  the  number  of  eggs  and  the  cost 
at  2  cents  each  is  represented  by  the  equation  c  =  2n.  In 
these  two  examples  /  and  c  are  the  same  function  of  n. 

The  symbol  f(x)  is  used  to  represent  a  function  of  x. 
It  is  read  function  of  x,  or  briefly,  /  of  x.  To  distinguish 
between  different  functions  of  x  other  letters  are  used,  as 
in  g(x)  or  F(x)* 

Thus,  if  the  function  3z+2  is  denoted  by  f(x)  and  if, 
in  the  same  discussion,  we  wish  to  refer  briefly  to  some 
other  function,  as  1^16  —  x2,  we  may- denote  the  latter  by 
the  symbol  F(x). 

6.  Evaluation  f  of  functions.  To  find  the  value  of  a 
function,  as  z2+5z-}-3,  for  a  given  value  of  x,  the  variable 
x  in  the  function  is  replaced  by  the  given  value.  If 
x2+5x+3  is  denoted  by  f(x),  then  32+5  •  3+3  is  denoted 

*  The  word  "function"  was  first  used  in  a  mathematical  sense 
in  1694  by  Leibnitz,  though  in  a  different  sense  from  that  given  here. 
In  October  of  the  same  year  James  Bernoulli  employed  the  word 
in  the  Leibnitzian  sense.  John  Bernoulli  employed  the  word  in  its 
modern  sense  in  a  letter  to  Leibnitz  in  June,  1698.  Leibnitz' 
answer  at  the  end  of  July  of  the  same  year  shows  that  he  too  had 
given  the  word  "function "  its  present  meaning.  The  new  technical 
term  was  first  employed  in  print  in  a  pamphlet  by  John  Bernoulli  in 
1706.  The  latter  was  also  the  first  to  define  the  word.  •  This  he 
did  in  the  Reports  of  the  French  Academy  of  1718. 

John  Bernoulli  and  Leibnitz  both  used  special  symbols  for 
"function "  in  1698.  Neither  used  the  symbol  defined  here.  Euler, 
in  a  scientific  publication  of  1734-35,  was  the  first  to  use  the  let- 
ter /,  followed  by  the  variable  inside  of  parentheses,  as  the  symbol 
for  function.  The  French  mathematician  Clairaut  was  at  the 
same  time  using  a  symbol  in  which  the  /  of  Euler's  symbol  was 
replaced  by  a  Greek  capital  letter  (Tropfke,  Geschichte  der  Elementar- 
Mathematik,  Band  I,  S.  142-43). 

f  Evaluation  means  to  find  the  value. 


4  THIRD-YEAR  MATHEMATICS 

by  /(3) .  Thus,  /(3)  is  the  result  obtained  by  replacing  x  in 
fix)  by  3,  or  the  functional  value  of  the  function  fix)  for 
the  particular  value  z=3  of  the  variable  x. 

EXERCISES 

1.  If  f(x)  =z2+3z+5,  find /(2),  /(0),  /(- 1),  /(a). 

/(2)=22+3  -2+5  =  15 
/(0)=02+3  -0+5=5 
/(-l)  =  (-l)2+3(-l)+5=3 
/(a)=a2+3a+5 

2.  If  fix)=x>-kc+Z  and  F(,x)  =  2x*-5,  find  F(6)-/(2). 

F(6)=2  -62-5  =  67 
/(2)=22-4-2+3  =  -l 
.'.  F(6)-/(2)=68 

3.  If/(2/)=2/3-3^+72/-l,  find/(l),/(-2),/(0). 

4.  If  /(r)  =  mr*+nr+p,  find /(-3),  /(J),  /(a). 

5.  If  fix)  =x2+2x+5  and  gix)  =z2-3a;+2,  find/(2)  +g(-l) ; 

find^3)' 

Linear  Function 

7.  The  function  ax+b.  Functions  like  2x+5,  \x— 7, 
3z+J,  are  of  the  form  ax+b,  where  a  and  6  are  constants 
and  a;  is  a  variable. 

EXERCISES 

1.  Which  of  the  following  functions  are  of  the  form  ax+b? 
|?+32;  2z2-4;  3(K;  v0+gt;  i-  j?(F-32);  2*-r; 
cos  x;  5*. 

2.  Give  other  examples  of  functions  of  the  form  ax+b. 

8.  Graph*  of  the  function  ax+b.  The  relation  be- 
tween the  variable  x  and  the  function  ax+b  may  be 
represented  graphically 

*  Graphing  was  introduced  as  a  systematic  mathematical 
method  by  Descartes  in  1637. 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN 


For  example,  a  boy,  on  a  certain  date,  deposits  in  a 
bank  the  sum  of  $3.  He  then  deposits  $2  regularly  at  the 
end  of  every  week.     How  » 

much  money  will  he  have 
in  x  weeks  ? 

Show  that  F(x)  =  2x+Z. 

Find  a  number  of  corre- 
sponding values  of  x  and 
F(x)  and  tabulate  them  as 
in  Fig.  1.  Plotting  the 
values  of  the  variable  x  as 
abscissas  and  the  values  of 
the  function  F(x)  as  ordi- 
nates,  we  obtain  the  straight 
line  AB. 


■FKxr        -j- 

7_ 

) 

J 

'10       ~) 

7~ 

jj 

/ 

7 

5L 

7 

~t~ 

) 

T_ 

7  '                ~x 

_1                     {_           Q 

~1 

f                                          X    F(X 

7               WT' 

/                -5                   15 

7                                                    2    7 

t                                                   3    9 

7                              4TT 

^^                 .•ifc 

42             3§[__   6IH 

Fig.  1 


/(*) 


9.  Linear  function.  The 

function  aa;+ b  is  a  function 

of  the  first  degree  in  z.     It 

is  also  called  a  linear  function  of  x,  because  the  graph  is  a 

straight  line. 

The  following  shows  that,  in  general,  a  straight  line  in 
a  plane  can  be  represented  by  an 
equation  of  the  form /(a;)  =  mx+b: 

1.  Let  P  be  any  point  on  the 
straight  line  ABC,  not  passing 
through  the  origin  0,  Fig.  2. 

Then  OQ  =  x  and  PQ=f(x). 

Denote  the  distances  AO  and 
BO  by  a  and  b,  respectively. 

Draw  BD±PQ  and  denote 
angle  DBP  by  the  letter  s. 


Fig.  2 


Show  that  tan  s 


DP 
BD 


J(x)-b 

X 


THIRD-YEAR  MATHEMATICS 

Denoting  the  value  of  tan  s  by  m,  we  have 

ifcO 


fix)  -b 

X 


.'.f(x)=mx+b 

2.  When  A B  passes  through  0,    

Fig-  3,  A/ 

x  x 

.'.f(x)=mx  FlG-3 

3.  When  a  line  is  parallel  to  the  x-axis  or  to  the  f(z)- 
axis,  show  that  its  equation  is  of  the  form  f(x)=c,  or  of 
the  form  x  =  c,  respectively. 

10.  Intercepts.  The  numbers  a  and  b,  Fig.  2,  are 
the  intercepts  made  by  the  line  A  C  on  the  z-axis  and 
the  F(x)-a,xk,  respectively. 

EXERCISES 

Construct  the  graphs  of  the  linear  functions  in  the  following 
examples: 

1.  The  length  of  a  circle  is  approximately  equal  to  3.14 
multiplied  by  the  diameter,  i.e.,  c  =  3.  Ud. 

2.  The  temperature  in  Fahrenheit  degrees  is  32  degrees 
greater  than  £  of  the  temperature  in  Centigrade  degrees   i  e 
F  =  J-(7+32.  '       ' 

3.  When  a  body  falls  from  rest,  the  velocity,  v,  at  any  time, 
t,  is  given  by  the  equation  v=gt,  where  0  =  32,  approximately. 

4.  Graph  the  equations  f(x)  =2z+3;  f(x)  =  -4;  x  =  2. 

Direct  Variation 

11.  Direct  variation.  In  preceding  work  (§200, 
Second-Year  Mathematics)  we  have  seen  that  the  state- 
ments y  varies  as  x,  y  is  directly  proportional  to  x,  or  y 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN        7 

varies  directly  as  x  are  expressed  algebraically  by  means 
of  the  equation  y  =  ex.  Thus,  the  statements :  A  man's 
pay  is  directly  proportional  to  the  number  of  days  he 
works,  the  distance  of  a  body  moving  at  a  uniform  rate 
varies  directly  as  the  time,  the  length  of  a  circle  varies 
as  the  radius,  are  written  respectively  p  =  ct,  d  =  ct,  l  =  cr. 
The  constant  c  is  the  constant  of  variation.  The  vari- 
ables p,  d,  and  I  are  linear  functions  of  t}  t,  and  r,  respec- 
tively. 

EXERCISES 

Express  the  following  statements  in  the  form  of  equations: 

1.  The  area  of  a  sphere  varies  as  the  square  of  the  radius. 

2.  The  volume  of  a  cylinder  varies  directly  as  the  square  of 
the  diameter  of  the  base  if  the  altitude  remains  constant. 

3.  The  area  of  an  equilateral  triangle  varies  as  the  square  of 
a  side. 

4.  When  a  body  falls. from  rest  (in  a  vacuum),  the  velocity 
varies  directly  as  the  time  of  falling. 

5.  When  a  spring  is  stretched  by  a  force,  /,  the  distance  the 
spring  is  stretched  (elongation)  varies  as  the  force  (Hooke's  law) . 

6.  The  average  consumption  of  coal  for  steam  boilers  varies 
directly  as  the  number  of  square  feet  of  grate  surface. 

7.  The  diagonal  of  a  cube  varies  as  the  edge.  The  edge  is 
5  when  the  diagonal  is  8.5.  Find  the  diagonal  when  the  edge 
is  10. 

1.  Show  that  d  =  c  •  e. 

2.  Determine  the  constant  of  variation : 

8  5 
Since  d  =  c  •  e,  8. 5  =  c  •  5;  .*.  c  =  -^-  =  1.7. 

3.  The  diagonal  may  now  be  determined :  d  =  (1 . 7)  (10)  =  17. 

8.  Represent  graphically  the  change  in  the  diagonal,  exer- 
cise 7,  as  the  edge  changes. 


8  THIRD-YEAR  MATHEMATICS 

9.  The  area  of  a  circle  varies  as  the  square  of  the  radius  and 
the  area  is  113  sq.  ft.  when  the  radius  is  6  feet.  Find  the  area 
of  a  circle  whose  radius  is  2\  feet. 

10.  The  time  required  by  a  pendulum  to  make  one  vibration 
varies  as  the  square  of  the  length.  A  pendulum  100  cm.  long 
vibrates  once  in  a  second.  •  What  is  the  time  of  vibration  of  a 
pendulum  49  cm.  long  ? 

11.  The  weight  of  a  liquid  is  directly  proportional  to  the 
volume.  If  10  cu.  ft.  of  water  weigh  625  lb.,  what  is  the  weight 
of  25  cubic  feet  ? 

12.  The  pressure  in  pounds  per  square  inch  of  a  column  of 
water  varies  directly  as  the  height  of  the  column  in  feet.  A 
column  of  water  2.5  ft.  high  exerts  a  pressure  of  1  08  lb  per 
square  inch.  Find  the  height  of  a  column  exerting  1  84 
pounds. 

Quadratic  Function 

12.  Quadratic  function.     The  functions  tit2,  s0+±gt2, 

3x2-4:X+2,  are  of  the  second  degree.     Show  that  they 

are    of  the   form   ax2+bx+c.     Functions   of   the   form 

ax2+bx+c  in  which  a^0  are  called  quadratic  functions. 

EXERCISES 

Show  that  the  following  functions  are  of  the  form  ax2+bx+c 
and  determine  in  each  case  the  values  of  a,  b,  and  c. 
1.  2-f  4x2-x  4.  £2_2 

2+5*     x      5  6.  _ 

3.  ax2-mx+2x2-12  6.  4  sin2  x+3  sin  x-7 

13.  Graph  of  the  function  ax2+bx+c.     In  §  9  it  was 

shown  that  the  graph  of  a  linear  function  is  a  straight  line. 
It  will  be  seen  that  the  graph  of  the  quadratic  function 
ax2+bx+c  is  a  smooth  curve  no  three  points  of  which  lie 
on  the  same  straight  line. 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN 


EXERCISES 

1.  Graph  the  function  f(x)  =  x2. 

Tabulate  values  of  f(x)  corresponding  to  values  of  x  between  —3 
and  +3  (Fig.  4). 

By  plotting  the  points 
corresponding  to  the  pairs 
of  numbers  in  the  table 
and  drawing  the  curve 
passing  through  these 
points,  the  graph  of  x2  is 
obtained.  This  curve  is 
called  a  parabola. 

If  #  is  a  negative  num- 
ber and  very  large,  f(x)  is 
positive  and  very  large, 

e.g.,  for  x=  —1,000,  f(x)  =  +1,000,000.  As  x  increases,  remaining 
negative,  f(x)  decreases.  As  x  approaches  zero,  f(x)  also  approaches 
zero.  As  x  continues  to  increase  indefinitely,  fix)  increases  without 
bound.     This  may  be  represented  in  a  table  as  follows: 


/(»•; 

X 

fix) 

T  Xjc 

_.. 

—3 

9 

v 

"    7 

3 

f 

—  1 

L 

5         ~i 

+  1 
+2 
+3 

1         

T, 

L. 

~J_J 

\ 

f~ 

nV 

(   z.iJX,^ 

^kv\ 

-3    -2    -      O 

+      \\2     t3  X 

X      - 

n 

Fig.  4 


X 

-00* 

negative,  increasing 

0 

positive,  increasing 

+  00 

fix) 

+  oof 

positive,  decreasing 

0 

positive,  increasing 

+  00 

*The  symbol  —  oo  means  increasing  without  bound  in  the  negative 
direction. 

t  The  symbol  +  oo  means  increasing  without  bound  in  the  positive 
direction. 

The  value  x  =  0  is  said  to  be  a  zero  of  /(x),  i.e.,  it  is  a  value  of  x 
such  that  the  corresponding  value  of  f(x)  is  zero. 

2.  Graph  the  positive  side  of  the  curve /(x)  =z2,  using  on  the 
z-axis  a  unit  equal  to  2  cm.,  and  on  the  f(x) -axis  a  unit  equal  to 
■§■  centimeter. 

Show  how  this  graph  may  be  used  as  a  device  for  finding  square 
roots  of  numbers. 

3.  Show  that  the  graph  of  the  f unction  f(x)  =  x2  is  symmetric 
with  respect  to  the  /(z)-axis. 

Show  that  the  /(x)-axis  is  the  perpendicular  bisector  of  the 
line-segment  joining  any  two  points  on  the  graph  which  have  equal 
ordinates. 


10 


THIRD-YEAR  MATHEMATICS 


4.  Graph  the  function  ttt2,  taking  tt  =  3.  14. 

5.  The  velocity  of  a  ball  thrown  vertically  upward  with  an 
initial  velocity  of 

64  ft.  per  second  is 
given  by  the  formula 
fl2  =  642-64A,  where  A 
is  the  height  attained 
at  any  time.  Show 
that  H  is  a  quadratic 
function  of  v.  Repre- 
sent graphically  the 
function  h=f(v),  for 
values  of  v  varying 
from  64  to  0. 

6.  Graph    the 

function  -,  \       .     _ 

f(x)  =  x2 —2x  -3. 

Compute  the  values  of  f(x)  as  in  the  table,  Fig.  5. 
Plot  the  points,  as  in  Fig.  5,  and  draw  the  graph. 
What  are  the  zeros  of  the  function  f(x)=x2-2x-3? 
Show  how   the  graph   may  be  used   to   solve  the  equation 
z2-3a;-3=0. 

Find  the  axis  of  symmetry  of  the  function  x2—2x—Z. 

^  Graph  the  following  functions  and  in  each,  case  locate  the 
axis  of  symmetry : 

7.  x2-6x+5  9.  x2+4x 

8.  3z2-llz-4  10.   -x2+6x-5 
Solve  graphically  the  following  quadratic  equations: 

11.  x2+4z+2  =  0  13.  z2+a._6  =  0 

12.  x2-5x+4:  =  0  14.  4-5x-x2  =  Q 

Graphical  Solution  of  Equations  of  Degree  Higher 
than  the  Second 

14.  Graph  of  a  cubic  function.  Functions  like  x3— x, 
x3-6z2+llz-6,  2z3-3z+2  are  functions  of  the  third 
degree  or  cubic  functions. 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN      11 


EXERCISES 

1.  Graph  the  function  f(x)=  x3  — x  and  locate  the  zeros. 


Plot  the  points  corresponding  to  the 
Draw 
these    . 


Compute  the  table,  Fig.  6. 
pairs  of  numbers  in  the  table, 
the    curve    passing    through 
points. 

What -are  the  zeros  of  x3  — xf 
State  how  the  graph  may  be  used 
to  solve  the  equation  x3  — x  =  0. 

2.  Graph  the  function 
f{x)  =  xz  —  6x2+ 1  lx  —  6 

and  use  the  graph  to  illustrate  the 
roots  of  the  equation  x3  —  6z2+Hz  —  6  =  0. 

3.  The  volume,  v,  of  a  sphere  is  ^-n-d3.  f 
Show  by  means  of  a  graph  the  changes  *-L 
of  the  volume  as  the  diameter  changes. 


-3 

-2 
-1 
1 
2 
0 
1 
2 
1 
2 
3 


-24 

-    6 

0 

4-1 

0 
3 
~8 
0 
b 
24 


I 


'fix 


m 


-10 


I 


Fig.  6 


15.  Graphical  solution  of  equations  of  degree  higher 
than  the  second.  Exercises  1  and  2,  §  14,  indicate  how 
the  graph  may  be  used  to  solve  some  cubic  equations. 
Equations  of  degree  higher  than  the  third  may  be  solved 
in  a  similar  way. 


EXERCISES 

Solve  the  following 
equations,  giving  the 
values  of  the  roots 
approximately  to  the 
first  decimal  place: 

1.  10x3+29x2-5x-6  =  0 
Graph  the  function 

f(x)  =  lux3  +29x2  -  5x  -  6, 

Fig.  7,  and  from  the  graph  determine 
approximately  the  required  values  of  x. 

2.  x3-2x2-7x-4  =  0 


X 

/(*) 

-4 

-162 

-3 

0 

-2 

+   40 

-1 

+  18 

0 

-     6 

1 

? 

0 

28 

2 

180 

Fig.  7 


12  THIRD-YEAR  MATHEMATICS 

Synthetic  Division.    Remainder  Theorem 

16.  The  work  of  evaluating  functions  for  different 
values  of  x  is  greatly  simplified  by  means  of  a  process 
called  synthetic  division  and  by  the  use  of  a  theorem  called 
the  remainder  theorem. 

17.  Synthetic  division.  The  process  of  synthetic  divi- 
sion is  illustrated  in  the  following  example: 

Divide  2x3-7x2-Sx+5  by  x-2. 

The  process  of  long  division  is  as  follows: 

2x*-r7x2-Zx+b  |  x-2 

2s3 -4s2  2x2-3z-9 

-3z2-3z 

-3z2+6z 


-9z+5 
-9x+18 


-13 


To  shorten  the  work  of  this  division,  omit  the  various 
powers  of  x,  writing  only  the  coefficients.  The  work  is 
now  arranged  as  follows: 


2- 

-7-3+5|  1- 

-2 

2- 

-4                2- 

-3- 

-9 

-3-3 

-3+6 

-9+5 

-9+18 

-13 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN     13 

Next,  omit  bringing  down  the  terms  of  the  dividend  as 
parts  of  the  remainders.     This  gives 

2-7-3+5  I  1-2 


2-4  2-3-9 


-3 

-3+6 


-9 
-9+18 


13 


The  coefficients  of  the  quotient  may  be  omitted,  as  they 
are  equal,  respectively,  to  the  first  coefficients  of  the  divi- 
dend and  of  the  remainders.  Similarly,  the  first  coeffi- 
cients of  the  partial  products  may  be  omitted.  Finally, 
since  the  first  coefficient  of  the  divisor  is  always  1,  it  need 
not  be  written.     This  reduces  the  process  to  the  following : 

2-7-3+5] ~2 
-4 


-3 

+6 


-9 
+  18 
-13 

In  the  process  of  division,  as  given  above,  the  partial 
products  —4,  +6,  and  +18  are  subtracted.  By  chan- 
ging the  —2  in  the  divisor  to  +2,  and  thus  changing 
the   signs  in   the   partial   products,  the  subtraction  of 


14  THIRD-YEAR  MATHEMATICS 

the  partial  product  is  changed  to  addition.     This  gives 
products, 

2-7-3+5  |_2_ 
4 


-3 
-6 


-18 


-13 

Finally,  the  work  may  be  condensed  into  the  following 
form: 

2-7-3+5  |_2_ 
4-6-18       "    . 


2-3-9-13 


Notice  that  the  first  three  successive  terms  in  the  lowest 
line  are  the  coefficients  of  the  quotient  and  the  last  term  is 
the  remainder.  Division  in  this  abbreviated  form  is 
called  synthetic  division. 

18.  Rule  for  synthetic  division.  To  divide  f(x)  by 
x  —  a,  f(x)  is  arranged  in  descending  powers  of  x,  supplying 
zeros  as  coefficients  of  missing  terms. 

The  coefficients  are  written  horizontally  and  the  first 
coefficient  is  brought  down. 

This  coefficient  is  multiplied  by  a  and  the  product  added 
to  the  second  coefficient. 

This  process  is  repeated  until  a  product  has  been  added 
to  the  last  coefficient. 

The  last  sum  is  the  remainder.  The  preceding  sums 
are  the  coefficients  ofxin  the  quotient,  arranged  in  descending 
order. 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN     15 

EXERCISES 

Divide  synthetically 

1.  z4-3z3+4z+2byz-3 

2.  3z3-4z+7by:c-l 

3.  5x3+2x2-3byz-5 

4.  4:x3+x2-3x-l  by  x+2 
Change  x+2  to  x-{-2). 

6.  2x4+6x-5byz+l 

19.  Remainder  theorem.  The  following  exercises 
show  that  the  value  of  f(x)  for  x  —  a  may  be  found  by 
dividing  f(x)  by  x—a: 

EXERCISES 

1.  Divide  f(x)  =  z2-6x+3  by  z-2. 

By  synthetic  division  the  quotient  and  remainder  are  obtained 
as  follows: 

1-6+3  \2_ 

2-8 


1-4-5 

Hence  the  quotient  is  x— 4  and  the  remainder  —5. 

2.  Find  the  value  of  f(x)  =  x2-6z+3  for  x  =  2. 
By  substitution  /(2)  =22-6  •  2+3=  -5. 

Thus  the  value  of  f(x)  ior  x  =  2  and  the  remainder  ob- 
tained by  dividing  /(x)  by  x-2  are  the  same. 
If  f(x)  denotes  a  function  of  x  in  the  form 
axn+bxn-l+cxn~2  .... 

the  result  of  exercises  1  and  2  may  be  stated  as  a  theorem 
as  follows: 

Whenf(x)  is  divided  by  x—a,  the  remainder  isf(a). 

This  principle  is  called  the  remainder  theorem. 


16  THIRD-YEAR  MATHEMATICS 

20.  Proof  of  the  remainder  theorem.  Since  divi- 
dend s  divisor  X  quotient  +  remainder,  it  follows  that 
f(x)  =  (x-a)Q(x)+R,  where  the  function  Q(x)  is  the 
quotient  and  the  constant  R  the  remainder. 

Substituting  a  for  x,     f(a)  =  (a  —  a)Q(a)  +R. 
.'.f(a)mQ.Q(a)+R. 
.'.f(a)  =  R,  which  was  to  be  proved. 

21.  Evaluation  of  /(*).  According  to  the  remainder 
theorem  the  value  off(x),  for  x  =  a,  may  be  found  by  dividing 
f(x)  by  x— a. 

EXERCISES 

Find  the  values  of  the  following  functions,  letting  x  take 
all  integral  values  from  —4  to  +3: 

1.  10x3+29x2-5x-6 
Dividing  synthetically  by  —4, 

10+29-  5-     6  |  -4 
-40+44-156 


162 


10-11+39 

.'./(-4)  =  -162 
Similarly,  find/(-3),/(-2),  etc. 

2.  x*-2x*-7x-±  3.  3z3-4z+7 

Solve  the  following  equations  graphically: 

4.  x3-kc2-2x+8  =  Q  6.  z3+x2-10x-10=0 

6.  a^-3x2-a;+3  =  0  7.  z3+3z2+2z=0 

Equations  of  Degree  Higher  than  the  Second 
Solved  by  Factoring 

22.  Factor  theorem.    In  §  19  it  was  shown  that  the 
remainder  obtained  by  dividing  f(x)   by  x—a  is  /(a). 
Hence,  if  the  remainder, /(a),  is  zero,  the  division  of  f(x) 
by  z  — a  is  exact  and  x—a  is  a  factor  of  f(x).    Thus 
x—a  is  a  factor  of  f{x)  if  f{a)  =  0. 

This  principle  is  called  the  factor  theorem. 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN     17 
EXERCISES 

The  principle  in  §  22  may  be  used  to  factor  the  following 
polynomials: 

1.  x3+2x2-9z-18 

If  the  polynomial  has  factors  of  the  form  x—a,  then  a  must  be  a 
divisor  of  18.  Thus  we  have  the  following  possibilities  for  a:  =*=1, 
±2,  ±3,  =*=6,  =*=9,  ±18. 

To  find/(l),  divide  synthetically  by  1, 

1     +2     -9     -18  [_1_ 

1+3-6 
1     +3     -6     -24 

Since /(l)  =  —24,  it  follows  that  x  —  1  is  not  a  factor.    Why  not  ? 

Similarly,  we  find 

/(  — 1)  =  —8  .'.  x+1  is  not  a  factor. 
/(2)  =  -20  .*.  3-2  is  not  a  factor. 

/(-2)  =     0  .-.  x+2  is  a  factor.     Why? 

Show  that  the  other  factor  is  x2— 9,  which  may  be  factored  as 
the  difference  of  two  squares. 

2.  tf-faP+llx-Q 

Since  there  are  no  powers  of  x  missing,  and  since  the  signs  are 
alternately  +  and  — ,  fix)  cannot  be  0  for  negative  values  of  x. 
Why? 

Hence  the  only  values  to  be  tried  are  1,  2,  3,  and  6. 

3.  z3+6x2+  llx+6 

Show  that/(:r)  ?^0  for  positive  values  of  x. 

4.  2^— 5i/3+52/2+52/— 6 

6.  Show  that  x — y  is  a  factor  of  xn — yn  if  n  is  an  integer,  and 
find  the  factors  of  x5—y5. 

7.  Show  that  x+y  is  a  factor  of  xn— yn  if  n  is  an  even  integer, 
and  factor  x*—ys. 

8.  Show  that  x-\-y  is  a  factor  of  xn-\-yn  if  n  is  an  odd  integer, 
and  factor  xb+yh» 


18  THIRD-YEAR  MATHEMATICS 

9.  Show  that  for  even  values  of  n  the  function  xn\yn  has 
no  factors  of  the  form  x+y  or  x— y. 

Solve  the  following  equations: 

10.  x*-7x+S=0 

The  factors  are  (x  —  1),  (x—2),  and  (x +3). 
.'.  (z-l)(z-2)(x+3)=0. 

x-l=0j 

x—  2=0  >  satisfy  the  given  equation. 
[z+3=0j 

[*i«lt 
Hence  jz2=2 

|x3=-3. 

11.  ^-192/-30  =  0  16.  ^-4rc-&c2+32  =  0 

12.  x3-5x-2=Q  17.  yz-y2=§y 

13.  *3-3H-2=0  18.  ^-1  =  0 

14.  2y*-y2-5y-2  =  0  Notice  that  the  roots  of 

this  equation  are  the  three 

15.  y2+\+y+l=:4:  cube  roots  of  1. 

y  yl*    u        y 

23.  Equations   of   degree   higher   than   the   second 
solved  like  quadratics. 


EXERCISES 

Solve  the  following  equations: 

1.  (x+2)2+3(z+2)  =  18 
By  factoring,  we  have 

(x+2-f6)(z+2-3)=0 
.     (*i=-8 

"     1*2  =  1 

2.  y*+2y*-80  =  0 

3.  (5i/-4)2-2(5?/-4)-63  =  0 

t  A  letter  with  a  subscript,  as  Xi,  is  read  "x  sub  one,"  or  "x  one." 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN      19 

4.  (x2-6:r)2+5(z2-6:c+20)-136  =  0 

Change  the  equation  to  the  form 

(x2-6z)2+5(z2-6z)+100-136=0, 
or  (z2-6z)2+5(x2-6:c)-36  =  0 

5.  3(7/+3rFl)2-7(?/2+3?/+l)+4  =  0 

6.  y2+y-2 — ^-  =  0 

*  y2+y 

7.  2G/2-3)+^|3+17  =  0 


The  Function  - 
x 

24.  Inverse  variation.     The  equivalent  statements  y 
varies  inversely  as  x  and  y  is  inversely  proportional  to  x 

are  expressed  algebraically  in  the  form  y=  .    Thus  the 


statement,  the  force  of  gravity  varies  inversely  as  the 
square  of  the  distance,  is  written  i=~rr  In  this  equa- 
tion c  is  the  constant  of  variation  and  /  is  a  function  of  d. 


EXERCISES 

Express  the  following  statements  by  means  of  equations: 

1.  The  number  of  vibrations  a  pendulum  makes  in  one 
second  varies  inversely  as  the  square  root  of  the  length. 

2.  The  amount  of  heat  received  from  a  stove  varies  inversely 
as  the  square  of  the  distance  from  it. 

3.  The  volume  of  gas  inclosed  in  a  cylinder  varies  inversely 
as  the  pressure. 

4.  The  pressure  which  a  given  quantity  of  air  at  constant 
temperature  exerts  against  the  walls  of  a  containing  vessel  is 
inversely  proportional  to  the  volume  occupied  (Boyle's  law). 


20 


THIRD-YEAR  MATHEMATICS 


Solve  the  following  problems: 

5.  The  intensity  of  light  on  an  object  varies  inversely  as 
the  square  of  the  distance  from  the  source  of  light  to  the  object. 
A  screen  15  ft.  from  a  lamp  is  moved  to  a  distance  of  5  ft.  from 
it.    How  much  does  this  increase  the  intensity  ? 

6.  A  pendulum  39.1  in.  long  makes  one  vibration  in  a 
second.  Find  the  length  of  a  pendulum  vibrating  4  times  a 
second. 

Use  exercise  1. 

7.  The  volume  of  a  gas  confined  in  a  cylinder  is  1  cu.  ft. 
when  the  pressure  is  5  pounds.  What  is  the  volume  when  the 
pressure  is  20  pounds  ? 

Use  exercise  4. 

25.  Graph  of  -.  Corresponding  values  of  x  and  f(x) 
for  c=  1  are  given  in  the  table,  Fig.  8. 


X 

/<*>                                                          10 

/(*) 

4 

i          

3 

I          . 

2 

i          

1 

1 

i 

2                                                                    5 

\      :    :  : 

4          

5 

:S:: 

6 

^ 

■3               ~±Z_                -1 

—  —  _^^ 

"'       '  »  f  J — '*,*.        a 

-I                   .2.      .        _Li 

^k 

—  i 

-6                                                          \ 

s 

-5                                                          H 

*~ 

-J 

-4                                                          "J* 

-i 

—2 

-1 

-1 

—2 

-1 

—3 

-4 

-1    ' 1 II II 1 1 II 1 II  Ml 

Fig.  8 


Plot  the  pairs  of  values  given  in  the  table,  Fig.  8. 
The  curve  obtained  consists  of  two  branches  which  do 
not  touch  either  axis.     Since  both  branches  are  obtained 


FUNCTIONS.     EQUATIONS  IN  ONE  UNKNOWN     21 

from  the  same  equation,  /(#)=-,  they  are  said  to  form 

x 

one  and  the  same  curve.     This  curve  is  called  a  hyperbola. 

Show  that  -  decreases  if  x  increases. 

x  x 

By  taking  x  large  enough,  -  can  be  made  smaller  than 
any  number  whatsoever.  1 

If  x  is  positive  and  decreases,  -  increases.     By  taking 

x  small  enough,  -  can  be  made  larger  than  any  number 
x 

however  great.     Often  this  fact  is  expressed  briefly  by 

the  statement  ^=  oo.    However,  this  does  not  mean  that 

1 

1  divided  by  0  has  a  value.     It  means  that  -  increases 

without  bound  as  x  approaches  zero  as  a  limit. 


EXERCISES 

12 

1.  Graph  the  function  f(x)  = — . 

x 

g 

2.  Graph  the  function  f(x)  =  - . 

Q 

3.  By  means  of  the  graph  of  the  function  -  find  a  meaning 

x 

for  the  expression  ~. 

4.  Discuss  the  changes  of  -  as  x  changes  from  —  oo  to  +  oo. 

x 

26.  Joint  variation.  The  area  of  a  triangle  varies 
as  the  product  of  the  base  by  the  altitude.  It  is  said  to 
vary  jointly  as  the  base  and  altitude. 

If  a  train  moves  with  a  uniform  speed  the  distance 
varies  j  ointly  as  the  rate  and  time.  This  may  be  expressed 
algebraically  by  means  of  the  equation  d=rt,  d  denoting 
the  distance,  r  the  rate,  and  t  the  time. 

The  statement  y  varies  jointly  as  x  and  z  is  expressed 
in  symbols  by  means  of  the  equation  y  =  cxz. 


22  THIRD-YEAR  MATHEMATICS 

EXERCISES 

Express  by  an  equation  each  of  the  following  statements: 

1.  The  distance  passed  over  by  a  train  varies  jointly  as  the 
rate  and  time. 

2.  The  pressure  of  water  on  the  bottom  of  a  basin  in  which 
it  is  contained  varies  jointly  as  the  area  of  the  bottom  and  the 
depth  of  the  water. 

3.  The  pressure  of  wind  on  a  wall  varies  jointly  as  the  area 
of  the  surface  and  the  square  of  the  velocity  of  the  wind. 

4.  The  volume  of  a  cylinder  varies  jointly  as  the  area  of  the 
base  and  altitude.  Compare  the  volumes  of  two  cylinders 
whose  altitudes  are  in  the  ratio  1:2. 

ex 
27.  Direct  and  inverse  variation.     If  y  =  —  then  y 

z 

varies  directly  as  x  and  inversely  as  z. 

EXERCISES 

Express  by  an  equation  each  of  the  following  statements : 

1.  The  cost  of  posts  for  a  fence  varies  directly  as  the  length 
of  the  fence  and  inversely  as  the  distance  between  the  posts. 

2.  The  resistance  to  an  electric  current  varies  directly  as  the 
length  of  the  wire  and  inversely  as  the  cross-section. 

3.  The  current  furnished  by  different  galvanic  cells  is  directly 
proportional  to  the  electromotive  force  and  inversely  propor- 
tional to  the  resistance  of  the  circuit  (Ohm's  law). 

Solve  the  following  problems : 

4.  If  y  varies  directly  as  x  and  inversely  as  z,  and  if  y  =  14 
when  x  —  7  and  z  =  1,  find  y  when  x  =  84  and  z  =  6. 

5.  The  interest  on  a  sum  varies  jointly  as  the  rate  and  prin- 
cipal. If  in  a  certain  number  of  years  the  interest  on  $2,000 
at  5  per  cent  is  $400,  what  is  the  interest  on  a  principal  of  $2,500 
at  5^-  per  cent  in  the  same  time  ? 


FUNCTIONS.    EQUATIONS  IN  ONE  UNKNOWN     23 

Summary 

28.  The   following   exercises   summarize   the   terms, 
symbols,  and  processes  taught  in  this  chapter: 

1.  Give  the  meaning  of  the  following  terms: 
function  intercept 
variable  direct  variation 
constant  inverse  variation 
evaluation  of  a  function  joint  variation 

linear,    quadratic,  cubic,  direct  and  inverse  variation 

function  parabola 

zero  of  a  function  hyperbola 

abscissa,  ordinate  synthetic  division 

2.  Explain  the  meaning  of  the  following  symbols:    f(x), 

Fix),  -oo,  +oo. 

3.  Tell  how  to  make  the  graph  of  f(x),  if  f(x)  is  linear; 
quadratic;  cubic. 

4.  Explain  the  use  of  synthetic  division  in  evaluating  a 
function  of  x. 

5.  Explain  how  to  solve  equations  in  one  unknown  of  the 
second  degree,  or  higher, 

1.  By  means  of  the  graph 

2.  By  factoring 

6.  Show  that  a  straight  line  can  be  represented  by  an  equa- 
tion of  the  form  f(x)  =  mx+b. 

7.  State  and  prove  the  remainder  theorem. 

8.  State  the  factor  theorem. 

9.  Represent  graphically,  y  =  x2—2x.     (Wisconsin.*) 

10.  The  attraction  of  gravitation  at  points  outside  the  earth's 
surface  varies  inversely  as  the  square  of  the  distance  from  the 

*  (Wisconsin)  means:    taken  from  an  entrance  examination 
given  by  the  University  of  Wisconsin. 


24  THIRD-YEAR  MATHEMATICS 

earth's  center.  If  the  attraction  on  a  certain  body  is  9  lb. 
at  the  surface  of  the  earth,  at  what  altitude  above  the  surface 
would  the  attraction  on  the  same  body  be  reduced  to  4  pounds  ? 
(Take  the  radius  of  the  earth  as  4,000  miles.)     (Harvard.) 

11.  Lights  of  equal  brightness  are  placed  in  three  corners  of 
a  square  room.  Show  that  the  intensity  of  the  illumination 
at  the  fourth  corner  is  T5^  that  at  the  center  of  the  room.  Given 
that  the  intensity  of  the  illumination  varies  inversely  as  the 
square  of  the  distance  from  the  light.     (Harvard.) 


CHAPTER  II 
TRIGONOMETRIC  FUNCTIONS 

29.  Angles  in  general.  In  the  preceding  course*  the 
sine,  cosine,  and  tangent  of  acute  angles  were  defined. 
These  functions  have  been  used  in  the  solution  of  right 
triangles,  §  257,  S.-Y.M.  However,  in  the  general  triangle 
obtuse  angles  as  well  as  acute  angles  are  found.  To  work 
with  the  general  triangle  the  notion  of  the  trigonometric 
functions  of  an  angle  must  be  extended  to  include  angles 
that  are  greater  than  90°. 

30.  Angle  as  amount  of  rotation. 
The  angle  XOA,  Fig.  9,  is  considered 
as  generated  by  the  rotation  of  a  line 
from  OX  around  to  the  position  OA.  pIG  9 
The  line  OX  is  called  the  initial  side 

and   OA   the  terminal  side  of  the  angle.     If  the  line 
revolves  from  OX  in  the  counter-clockwise  direction,  the 
angle  XOA  is  positive ;  if  it  revolves  in 
the  clockwise  direction,  the  angle  formed 
is  negative. 


31.  Quadrants.    Two  lines  at  right  m 

angles,  Fig.  10,  divide  the  plane  around  Y' 

the  point  of  intersection,  0,  into  four  FlG   10 

equal   parts   called    quadrants.     The 
quadrants  are  numbered  as  follows:   XOY  is  the  first 
quadrant,  YOX'  the  second,  X'O  Yf  the  third,  and  Y'OX 
the  fourth. 

*  Second-Year  Mathematics,  §  248. 
25 


26 


THIRD-YEAR  MATHEMATICS 


An  angle  is  said  to  be  in  the  first,  second,  third,  or 
fourth   quadrant,   according  as  its 
terminal  side  lies  in  the  first,  second, 
third,  or  fourth  quadrant,  the  initial 
side  having  been  placed  on  OX. 

Hence  angles  between  0°  and  90°, 
90°  and  180°,  180°  and  270°,  and 
270°  and  360°  are  said  to  be  in  the 
first,  second,  third,  and  fourth  quad- 
rants, respectively  (Fig.  11). 


EXERCISES 

1.  In  which  quadrant  is  Z.XOA,  Fig.  11?  ZXOBf 
ZXOCf     ZXOD? 

2.  Draw  the  following  angles  and  in  each  case  state  the 
quadrant  in  which  the  angle  lies:  20°,  160°,  240°,  315°,  545°, 
-40°,  -220°. 

32.  Trigonometric  functions.  Let  XOA,  Figs.  12  to 
15,  be  a  given  angle.     From  any  point,  P,  of  the  terminal 


b       MX 


Fig.  12 


Fig.  14 


Fig.  13 


Fig.  15 


TRIGONOMETRIC  FUNCTIONS  27 

line  OA,  drop  a  perpendicular,  PM,  to  the  initial  line  OX 

(produced  if  necessary).     This  forms  a  right  triangle, 

MOP. 

The  sine  of  ZXOA  is  the  ratio  of  the  side  of  AMOP 

that  lies  opposite  the  vertex  0  to  the  hypotenuse,  i.e., 

MP .         , 

jyp  in  each  case. 

The  cosine  of   ZXOA  is  the  ratio  of  the  side  of 

AMOP   that    lies    adjacent   to   0  to   the   hypotenuse, 

.       OM 
i.e.,  Qp  . 

The  tangent  of  Z  XOA  is  the  ratio  of  the  side  opposite 

MP 

0  to  the  side  adjacent,  i.e.,  ^r? . 

The  cotangent  of  ZXOA  is  the  ratio  of  the  side 
adjacent  to  0  to  the  side  opposite  0,  i.e.,  TTp. 

The  secant  of  ZXOA  is  the  ratio  of  the  hypotenuse 

OP 

to  the  side  adjacent  to  0,  i.e.,  7^7  > 

The  cosecant  of  Z  XOA  is  the  ratio  of  the  hypotenuse 

OP 
to  the  side  opposite  0,  i.e.,  y^p . 

These  are  called  the  ratio-definitions  of  the  trigo- 
nometric functions. 

Suggest  why  these  ratios  are  called  functions. 

33.  Signs  of  the  functions  in  each  quadrant.     If  the 

side  opposite  to  the  vertex  extends  upward  from  the  initial 
line  OX  it  is  considered  positive,  if  it  extends  downward  it 
is  negative. 

If  the  adjacent  side  extends  to  the  right  of  0  it  is  posi- 
tive, if  to  the  left  it  is  negative. 

The  hypotenuse  is  always  regarded  as  positive. 


28 


THIRD-YEAR  MATHEMATICS 


Denoting  the  measure  of  angle  XOA  by  the  Greek 
letter  a  (alpha)  and  the  lengths  of  the  sides  of  AM  OP 
by  a,  b,  and  c,  respectively,  show  that  the  statements  of 
§  32  take  the  following  form : 


^Qeti52^~-~^_Quadrants 

I 

ii 

III 

IV 

Sine  a 

+-: 

+e 

a 
c 

a 
c 

Cosine  a 

i 

c 

_b 
c 

*\ 

Tangent  a 

+i 

a 
~b 

+s 

a 
~b 

Cotangent  a 

+5 

_b 
a 

«5 

_b 

a 

Secant  a 

+i 

c 
~b 

c 

b 

+1 

Cosecant  a 

+5 

a 

c 

a 

c 
a 

The  signs  of  the  functions  in  the  various  quadrants 
should  be  thoroughly  well  known.    The  following  diagrams, 


sine-cosecant 


cosine-secant 


tangent-cotangent 


Fig.  16 


Fig.    16,   will    be    helpful    in   remembering  the   correct 
algebraic  signs  of  the  various  functions, 


TRIGONOMETRIC  FUNCTIONS 


29 


34.  Values  of  the  trigonometric  functions  found 
by  means  of  a  drawing.  In  the  following  exer- 
cises construct  on 
squared  paper  the 
given  angle.  Draw 
the  defining  triangle 
MOP  as  in  Fig.  17. 
Measure  the  sides  of  +* 
the  triangle  and  deter- 
mine the  algebraic 
signs  and  the  numeri- 
cal values  of  the  func- 
tions of  the  given  angle. 


\A„ 

V 

s 

V 

\ 

s 

^   IT 

^,+  1.7 

Ml                 V 

\-£                     S 

\ 

V 

s 

51 

-Jt^v 

\_S 

M             -1-0         0                                   X 

Fig.  17 


EXERCISES 

1.  Find  the  value  of  the  sine  of  125°,  Fig.  17. 

2.  Find  the  values  of  the  functions  of  the  following  angles: 
140°,  220°,  245°,  315°. 

35.  Abbreviations*  of  the  names  of  the  functions. 

The  expressions  sine  of  angle  a,  cosine  of  angle  a,  tangent 
of  angle  a,  etc.,  are  usually  written  in  the  following  abbre- 
viated forms:  sin  a,  cos  a,  tan  a,  cot  a,  sec  a,  and  esc  a. 

36.  Given  the  value  of  one  function  of  an  angle  to 
construct  the  angle. 


EXERCISES 

1.  Given  tan  a  =  f.  Construct 
angle  a. 

Show  that  there  are  two  angles 
whose  tangent  is  f ,  one  in  the  first 
and  one  in  the  third  quadrant,  Fig.  18. 
Construct  these  angles  and  measure  them  with  a  protractor. 

*  See  note,  p.  138,  Second-Year  Mathematics. 


Fig.  18 


30 


THIRD-YEAR  MATHEMATICS 


2.  Construct  angles  A,  x  and  a,  having  given  cos  A  =  \, 
tan  x  =  —3,  and  cot  a=  V  3. 

3.  Construct  the  angles  x  and  y  given  by  tan  x  =  f ,  cos  2/  =  -§-. 

4.  Construct  angles  a,  having  given  cos  a=  —  |,  sin  a=  +|-, 
tan  a  =  3,  tan  a=  —3,  cot  a  =  4. 

37.  Inverse  functions.  According  to  §  36  an  angle 
may  be  found  if  the  value  of  one  of  the  trigonometric 
functions  is  known.  Thus,  the  equation  sin  x=^ 
determines  x  as  an  angle,  or  arc,*  whose  sine  is  §.  The 
statement  x  is  an  angle  whose  sine  is  y  is  usually  written 
briefly:  x  =  sin~1y  or  x  =  arc  sin  y,  read  inverse  sine  and 
arc  sine  respectively. 

Give  the  meaning  of  the  following : 

£  =  tan-13,  y  =  arc  cos  (  —  \),  A=&rc  sin  f. 

38.  Given  the  value  of  one  function  of  an  angle,  to 
determine  the  values  of  the  other  functions. 


EXERCISES 

1.  Given  tan  a=—  f,  a  being  in  the  second  quadrant,  find 
the  other  functions  of  a. 

Construct  AMOP,  Fig.  19, 
having  OM=-3,  MP  =  4.  Com- 
pute OP. 

2.  Let  cot  z  =  f.  To  find  the 
other  functions  of  x,  x  being  in  the 
third  quadrant. 

3.  Given  tan  x  =  \.  If  Z  is  in 
the  third  quadrant,  give  the  values 
of  the  other  functions. 


M 


Fig.  19 


*  Since  an  angle  whose  vertex  is  at  the  center  of  a  circle  has  the 
same  measure,  in  degrees,  as  the  intercepted  arc,  the  trigonometric 
functions  may  be  regarded  as  functions  of  the  arc  instead  of  the 
angle. 


TRIGONOMETRIC  FUNCTIONS 


31 


4.  Let  sin  a  =  ■£-§-,  find  cot  a,  Fig.  20. 

5.  If  Z  A  is  in  the  third  quadrant  and  tan  A  =  ^3 ,  find  sec  A 
and  sin  A. 

6.  Find  the  values  of  the  functions  of  z  if  sec  z  =  J . 


\*5 

25/ 

13 

16 

-vSoT 

v^ol 

Fig.  20 


Fig.  21 


7.  Find  cos  x  if  tan  x- 


2mn 
m2—ri' 


,  Fig.  21. 


8.  If  sin  a  = 


2  a 


1+a2 


find  cos  a. 


2#w 

9.  If  tan  a=  n   y  n ,  find  sin  a. 

z2— 2/2 


10.  If  #  =  arc  sin  y,  find  tan  x  in  terms  of  y. 


Changes  of  the  Trigonometric  Functions  as  the  Angle 
Changes  from  0°  to  360° 

39.  Trigonometric   functions   represented    by   lines. 

Since  the  trigonometric  functions  are  ratios  it  is  possible 
to  represent  them  graphically  by  means  of  line-segments. 
This  simplifies  greatly  the  study  of  the  changes  of 
the  functions  that  depend  upon  the  changes  of  the 
angle. 


32 


THIRD-YEAR  MATHEMATICS 


40.  Line    representation    of    the    sine    and    cosine 
functions.     Let  ZXOA,  Fig.  22,  be  any  angle. 

With  a  radius  equal  to  1  and  the 
center  at  0  draw  a  circle. 

A  circle  of  radius  1  is  called  a 
unitrcircle. 

From  the  point  of  intersection, 
P,  of  the  unit-circle  and  side  OA 
draw  PM±0X. 

Denoting  ZXOA  by  a,  we  have 
MP    MP 


sin  a  =  jyp-  =  — —  =  M P,  i.e. 


Fig.  22 

the  measure 


of  MP  is  the  same  as  sin  a.     The  segment  MP  represents 

sina'  OM    OM 

Similarly,  cos  a  =  jyp  =  — —  =  OM ,  i.e.,  the  measure 

of  OM  is  the  same  as  cos  a.     Hence,  OM  represents  cos  a. 


EXERCISE 

Draw  angles  lying  in  the  second,  third,  and  fourth  quadrants 
and  represent  by  line-segments  the  values  of  the  sine  and  the 
cosine  functions. 

41.  Changes  of  the  sine  and  the  cosine  functions.    In 
Fig.  23  MiPi,  M2P»,  etc.,  represent 
sin  a,  and  OM1}  OM2}  etc.,  represent 
cos  a. 

As  a  decreases,  sin  a  decreases  and 
cos  a  increases.  When  OP  coincides 
with  OX,  a  =  0,sin0°  =  0,andcos0°=  1. 

As  a  increases  from  0°  to  90°,  sin  a 
increases  and  cos  a  decreases,  both 
being  positive. 

When  a  =  90°,  MP  coincides  with  OP  and  sin  90°  -  -f  1, 
while  cos  90°  =  0. 


Fig.  23 


TRIGONOMETRIC  FUNCTIONS 


33 


/ 


As  a  increases  from  90°  to  180°,  sin  a  decreases  from 
1  to  0  and  cos  a  decreases  from  0  to  —  1,  Fig.  24. 

As  a  increases  from  180°  to  270°,  sin  a  decreases  from 
0  to  —  1  and  cos  a  increases  from  —  1  to  0,  Fig.  25. 


Fig.  24 


Fig.  25 


As  a  increases  from  270°  to  360°,  sin  a  increases  from 
—  1  to  0  and  cos  a  increases  from  0  to  1,  Fig.  26. 

The  following  table  gives  the  values  of  sin  a  and  cos  a 
for  special  values  of  a,  a  being  less  than,  or  at  most  equal 
to,  360°: 


„        ~~~~~~~~— __         Angle 

Function         ^ — -___^ 

0° 

30° 

45° 

60° 

90° 

180° 

270° 

360° 

Sine 

0 

1 
2 

l/2 
2. 

t/3 

2 

1 

0 

-1 

0 

Cosine 

1 

2 

V'2 
2 

1 
2 

0 

-1 

0 

1 

The  sine  and  the  cosine  functions  cannot  be  greater 
than  +1  and  not  less  than  —1.     Why? 


EXERCISES 

1.  Describe  the  variation  of  sin  a;  as  re  increases  from  0° 
to  360°.    Illustrate  by  means  of  a  figure. 

2.  Describe  the  variation  of  cos  x  as  x  increases  from  0°  to 
360°.    Illustrate  by  means  of  a  figure. 


34 


THIRD-YEAR  MATHEMATICS 


42.  Line  representation  of  the  tangent  and  secant 
functions.  Let  ZXOA,  Fig.  27,  be  any  angle.  With 
radius  equal  to  1  and  with  0  as 
center  draw  a  circle.  At  X  draw 
XT  tangent  to  circle  0. 

Denoting  Z  XOA  by  a,  we  have 

■A  A.      JLJx      -it  a    •  i 

tan  a  =  7TTF  =  — —  =  AA,  i.e.,  the  part 


OX       1 

of  the  tangent  at  X  intercepted  by 
the  initial  and  terminal  sides  of 
ZXOA  has  the  same  measure  as 
tan  a.    Hence  XA  represents  tan  a. 

a.     .,     ,  OA     OA     r.A  Fig.  27 

Similarly,  sec  a  =  -^  =  — -  =  OA , 

i.e..  the  measure  of  OA  is  the  same  as  the  value  of  sec  a. 
Hence,  sec  a  is  represented  by  OA. 

43.  Changes  of  the  tangent  and  secant  functions.    As 
a  decreases,  Fig.  28,  XA  decreases. 
For  a  =  0°,  XA=0,  i.e.,  tan  a  =  0. 

As  a  increases  from  0°  to  90°, 
tan  a  increases. 

As  a  approaches  nearer  and 
nearer  to  90°,  XA  increases  without 
bound.  Thus,  tan  a  has  no  definite 
value  when  a  =  90°.  Often  this  fact 
is  expressed  symbolically  by  the 
statement  tan  90°  =  +  a>.  How- 
ever, this  statement  does  not  mean 
that  90°  has  a  tangent.     It  means  pIG  28 

that  tan  a,  remaining  positive,  in- 
creases without  bound  as  a  approaches  90°  as  a  limit. 

Show  that  sec  a  increases  from  1  to  +  oo  as  a  changes 
from  0°  to  90°. 

Show  that  the  sign  of  sec  a,  Fig.  28,  is  the  same  as  the 
sign  of  cos  a. 


TRIGONOMETRIC  FUNCTIONS 


35 


When  a  lies  in  the  second  quad- 
rant, Fig.  29,  tan  a  is  represented  by 
the  part  of  the  tangent  at  X  which 
is  intercepted  between  the  initial 
side  OX  and  the  extension  of  the 
terminal  side  of  ZXOA.  The  fact 
that  XAi,  XA2,  etc.,  extend  down- 
ward from  OX  shows  that  tan  a  is 
negative  in  the  second  quadrant. 
When  a  lies  in  the  second  quadrant 
and  decreases  approaching  90°, 
tan  a  increases  without  bound, 
always  being  negative.  This  is 
expressed  in  symbols  by  means  of 
the  statement  tan  90°=  —  00. 
As  a  approaches   180°  tan  a  approaches  zero. 


Fig.  29 


A, 


A, 


EXERCISES 

1.  Show  that  sec  a  changes  from  —  00  to  —  1  as  a  changes 
from  90°  to  180°. 

2.  As  a  changes  from  180°  to  270°  show  that  tan  a  changes 
from  0  to  +  00 ;  that  sec  a  changes  from  —  1  to  —  00. 

3.  As  a  changes  from  270°  to  360°  show  that  tan  a  changes 
from  —  co  to  0;  that  sec  a  changes  from  +00  to  +1. 

The  following  table  gives  the  changes  of  tan  a  and  of 
sec  a  as  a  changes  from  0°  to  360° : 


^"~-\^     Angle 
Function    ^**\^^ 

0° 

90° 

180° 

270° 

360° 

Tangent 

0 

=t  00 

0 

=±=  GO 

0 

Secant 

+  1 

=t  CO 

-1 

=T=GO 

+  1 

36 


THIRD-YEAR  MATHEMATICS 


44.  Line  representation  of  the  cotangent  and  cosecant 
functions. 

Let  ZXOA,  Fig.  30,  be  any  angle. 


Fig.  30 

Draw  a  unit-circle  with  the  center  at  0.    At  Y  draw 

YT  tangent  to  circle  0. 

Show  that  Z  XOA  =  Z  YAO. 

Denote  ZXOA  by  a. 

YA     YA 
Show  that  cot  a  =  jyy  =  —r-  =  YA,  i.e.,  the  part  of  the 

tangent  at  Y  which  is  intercepted  by  OY  and  OA  has  the 
same  numerical  measure  as  cot  a. 

Similarly,  show  that  esc  a  =  ^Ty  =  ~ r~ =  ^ . 

Hence  OA  has  the  same  numerical  value  as  esc  a  and 
represents  esc  a. 

When  a  is  obtuse,  point  A  is  to  the  left  of  Y  and  there- 
fore YA  is  negative. 

When  a  is  in  the  third  quadrant,  show  that  YA  is 
positive. 

When  a  is  in  the  fourth  quadrant,  show  that  YA  is 
negative. 

EXERCISE 

Describe  the  variations  of  cot  a  and  esc  a  as  a  increases  from 
0°  to  360°.    Illustrate  by  means  of  figures. 


TRIGONOMETRIC  FUNCTIONS 


37 


45.  Table  giving  the  changes  of  the  functions  as  the 
angle  changes  from  0°  to  360°.  The  following  table  gives 
changes  of  functions  of  a  as  a  changes  from  0°  to  360° : 


^^\^  Angle 
Function"\^ 

0°  to  90° 

90°  to  180° 

180°  to  270° 

270°  to  360° 

Sine 

0   to  +1 

+1    to      0 

0   to  -1 

-1    to      0 

Cosecant 

+  oo  to  +1 

+  1    to  +00 

-  oo  to  - 1 

—  1    to  -  CO 

Cosine 

+  1    to      0 

0    to  -1 

-1    to      0 

0    to  +1 

Secant 

+  1    to  +  <* 

-oo  to  -a 

—  1    to  —oo 

+  oo  to  +1 

Tangent 

0    to  +  co 

-  oo  to      0 

0    to  +oo 

-co  to      0 

Cotangent. . .  . 

+  oo  to      0 

0    to  -oo 

+  oo  to      0 

0    to  -oo 

Graphs  of  the  Trigonometric  Functions 

46.  Radian  measure.  There  are  two  current  methods 
of  measuring  angles,  viz.,  degree  measure  and  radian,  or 
circular,  measure.  The  student  is  already  familiar  with 
the  first,  in  which  the  unit-angle  is 
a  degree  consisting  of  -g-J-jr  of  a 
complete  revolution.  Before  con- 
structing the  graphs  of  the  trigono- 
metric functions  we  will  examine 
the  second  method  and  its  advan- 
tages over  the  first. 

Let  a  be  the  measure,  in 
degrees,  of  Z.A0B,  Fig.  31.  Draw 
the  unit-circle  having  the  center  at  the  vertex  0. 

Since  a  is  measured  by  the  intercepted  arc,  the  length 
of  AB  may  be  used  to  express  the  value  of  a.  In  that  case 
the  unit  of  measure  is  an  arc  of  unit  length,  or  the  angle  inter- 
cepting an  arc  equal  in  length  to  the  radius  of  the  circle.     This 


Fig.  31 


38 


THIRD-YEAR  MATHEMATICS 


method  of  measuring  angles  is  called  radian  measurement, 
or  circular  measurement. 

47.  Radian.  The  unit  of  circular 
measure,  called  the  radian,  is  the 
angle  that  intercepts  an  arc  equal  in 
length  to  the  radius  of  the  circle, 
Fig.  32.  When  the  unit  of  measure 
is  not  indicated  it  is  understood  to  be 
a  radian.  Thus  LABC=\  means 
that  /.ABC  is  ^  of  a  radian. 

48.  Relation  between  circular  measure  and  degree 
measure.     Let  Z  BOA,  Fig.  32,  be  a  radian. 

Since  the  length  of  a  semicircumference  is  irr  and  since 
AB*  =  r,  it  follows  that 

7T  radians  =  180°, 
where  7r  =  3. 14159  approximately; 
180  V 


/ 


Fig.  32 


a  radian 


=  f  -J3f-  )  =  57° .  3  approximately. 


Show  that 


10=(iio)radians- 


Fig.  33 


49.  Relation  between  an  angle, 
the  intercepted  arc,  and  the  radius 
of  the  circle.  Let  the  Greek  letter  6 
itheta)  denote  the  number  of  radians 
in  a  given  angle,  Fig.  33. 

Then  T  =  -  ,  or  8  =  -  ,  or,  in  words : 
1     r'  r-      ' 

The  number  of  radians  in  an  angle  is  equal  to  the  length 
of  the  intercepted  arc  divided  by  the  radius  of  the  circle. 
Show  that         s=r8. 
Express  the  equation 

s  =  rd  in  words. 


*The  symbol  AB  means  arc  AB. 


TRIGONOMETRIC  FUNCTIONS  39 

EXERCISES 

1.  Express  10°  15'  in  circular  measure. 

10°  15'  =  (10 J)°  =  (10j)  (j^)  radians  =  .  19,  approximately. 

2.  Express  the  following  angles  in  circular  measure:    10°, 
8°  30',  -50°,  58°. 

3.  Find  the  number  of  degrees  in  an  angle  whose  circular 


5  radians  =  „(  —  j  =  76?39,  approximately. 
4.  Express  in  degrees  the  following  angles:  § ,  \,  .752,  3. 14. 


6.  Express  in  radians  the  following  angles:  0°,  30°,  45°,  60°, 
90°,  180°,  270°,  360°. 

6.  Express  the  following  angles  in  degree  measure:  -,  -,  -, 

O  TV      TT 

7.  Give  the  values  of  the  following  functions:  sin  ^,  tan  — , 

3tt  l  l 

COS  7T,  CSC  —  . 

8.  The  radius  of  a  circle  is  3  feet.  Find  the  length  of  the 
arc  intercepted  by  an  angle  at  the  center  equal  to  1  J. 

9.  The  radius  of  a  circle  is  10  feet.  What  is  the  length  of  an 
arc  intercepted  by  an  angle  of  80°  ? 

10.  What  is  the  circular  measure  of  an  angle  at  the  center 
of  a  circle  intercepting  an  arc  equal  to  §  of  the  radius  ? 

11.  Prove  that  the  area  of  a  sector  of  a  circle  is  — ,  r  being  the 
radius  and  0  the  angle  at  the  center,  in  radians. 

12.  Prove  that  the  area  of  a  segment  of  a  circle  is  — — — - — . 

Z  Z 

13.  The  radius  of  a  circle  is  10  feet.  Find  the  angle  at  the 
center  intercepting  an  arc  2  ft.  long.  Express  the  result  in 
degrees  and  in  radians. 


40 


THIRD-YEAR  MATHEMATICS 


50.  Graphing  the  trigonometric  functions.  To  graph 
the  trigonometric  functions  we  may  plot  the  correspond- 
ing values  of  angle  and  function  as  obtained  from  a  table 
of  functions.  However,  the  following  will  show  that  the 
line  representation  affords  a  very  simple  way  of  making 
the  graphs. 

51.  The  sine  curve.  Lay  off  on  OX,  Fig.  34,  to  a  con- 
venient  scale,    distances   representing    a  =  --,  -  . 

7T  12      6 

2  •  •  •  .  ir,  etc.,  where  tt  =  3.14.     At  the  points  thus 

obtained  lay  off  vertically  the  corresponding  distances 
representing  sin  a.  Draw  a  smooth  curve  through  the 
top  points  of  these  vertical  lines. 


Fig.  34 


This  is  the  sine  curve. 


EXERCISES 

From  a  study  of  Fig.  34  answer  the  following  questions: 

1.  As  a  varies  from  0  to  360  how  does  sin  a  vary  ? 

2.  At  what  places  is  the  change  in  sin  a  most  rapid  ?    Where 
is  the  change  slowest  ? 

3.  How  does  the  curve  show  that  the  sine  function  repeats 
its  values  at  intervals  of  2tt,  or  360°  ? 

4.  What  is  the  largest  value  of  sin  a  ?    What  is  the  smallest 
value  ? 


TRIGONOMETRIC  FUNCTIONS 


41 


52.  Periodic  function.  A  function  whose  values  are 
repeated  at  definite  intervals  as  the  variable  increases  is 
a  periodic  function. 


EXERCISES 

1.  Show  from  Fig.  34  that  the  period  of  the  sine  function  is 
2tt,  i.e.,  show  that  sin  (a+27r)  =sin  a. 

2.  Show  that  sin  (— a)  =  —  sin  a. 

3.  Show  that  sin  (■*— a)  =sin  a. 

These  exercises  suggest  certain  facts  about  the  sine 
function  to  be  proved  in  §§  58-63. 

53.  The  cosine  curve.    The  curve  in  Fig.  35  is  the 
cosine  curve.     To  construct  it  follow  the  directions  given 


s 

> 

- 

:z 

— T-= 

f 

/  / 

- 

-• 

■" 

s< 

/ 

/ 

/ 

- 

I 

// 

s 

_ 

\ 

r~\ 

) 

7T 

7 

" 

/.{7T 

2TT 

57T 

v 

\k  v 

\ 

- 

2 

~  i 

2 

\ 

\ 

\ 

T 

^. 

2a 

:^= 

- 

— 

_ 

^ 

d 

' 

Fig.  35 


for  the  construction  of  the  sine  curve,  §  51.  Notice  that 
the  cosine  curve  has  the  same  shape  as  the  sine  curve  and 
differs  from  it  only  in  position. 


EXERCISES 


Give  a  discussion  of  the  cosine  curve  similar  to  that  given 
for  the  sine  curve,  §§51  and  52. 


42 


THIRD-YEAR  MATHEMATICS 


54.  The  tangent  curve.    Draw  the  tangent  curve, 
Fig.  36,  and  give  a  discussion  as  in  §  51. 


Fig.  36 


55.  The  cotangent 
curve.  Using  Fig.  37, 
draw  the  cotangent 
curve,  Fig.  38,  and 
give  a  discussion  simi- 
lar to  that  given  for 
the  sine,  cosine,  and 
tangent  curves. 


I 


:fc 


y 


Fig.  37 


TRIGONOMETRIC  FUNCTIONS 


43 


Fig.  38 


56.  The  secant 
and  cosecant  curves. 
These  curves,  given  in 
Fig.  39,  may  be  con- 
structed as  in  §§  54 
and  55 .  The  solid  line 
represents  the  secant, 
the  dotted  line  the 
cosecant,  function. 


Fig.  39 


44 


THIRD-YEAR  MATHEMATICS 


The  Trigonometric  Functions  of  Negative  Angles 

57.  Positive  and  negative  angles.  By  rotating  AB, 
Fig.  40,  around  A  until  it  takes  the 
position  AC,  angle  BAC  is  formed. 
By  rotating  AB  in  the  opposite  direc- 
tion, angle  BAC  is  formed.  It  is 
customary  to  consider  an  angle  posi- 
tive when  it  is  formed  by  rotating  a  line 
counter-clockwise,  and  negative  when 
it  is  formed  by  clockwise  rotation.  Fig.  40 

58.  Trigonometric  functions  of  —a  in  terms  of  the 
functions  of  a.  Denoting  /.BAC,  Figs.  41  to  44,  by 
a,  then  ZBAC=-a. 


Fig.  42 


Fia.  44 


With  A  as  center  and  radius  equal  to  1  draw  CC. 

Draw  chord  CC.  

Then  AD  is  the  perpendicular  bisector  of  CC* 


*  CC  means  chord,  or  segment  CC 


TRIGONOMETRIC  FUNCTIONS  45 

Show  that  triangles  DAC  and  DAC,  Figs.  41  to  44, 
are  congruent. 

.-.ADAC^ADAC 

sin  a  =  DC 
sin  (-a)  =DC'=-DC=  -sin  a 
Similarly,  cos  ( —  a)  =  AD  —  cos  a 

sin  ( —  a)  _  —  sin  a 
cos  ( —  a)       cos  a 
.*.  tan  ( -  a)  =  -  tan  a        Why  ? 


EXERCISES 

1.  Show  that 

1.  cot  (—a)  =  —COt  a  3.     CSC  (—a)  =  — CSC  a 

2.  sec  (—  a)=  sec  a 

2.  Express  in  terms  of  functions  of  positive  angles  the 
values  of  all  functions  of  the  following  angles: 

-30°,  -45°,  -60°. 

The  exercises  in  §  58  show  that  any  trigonometric 
function  of  —a  is  equal  numerically  to  the  same  function  of  a, 
but  differs  in  sign,  with  the  exception  of  the  cosine  and  secant. 

"  ■  ■  ■  -  m 

59.  In  §  34  we  have  seen  that  the  values  of  the  trigo- 
nometric functions  of  any  angle  may  be  found  from  a  draw- 
ing. The  values  of  the  functions  of  angles  less  than  90° 
may  be  found  by  means  of  special  tables  (see  p.  356). 
The  values  of  the  functions  of  angles  greater  than  90°  can 
be  found  by  means  of  certain  relations  which  enable  us  to 
express  the  functions  of  any  angle  in  terms  of  some  func- 
tion of  an  angle  less  than  90°.  These  relations  are  to  be 
worked  out  in  the  following  sections. 


46 


THIRD-YEAR  MATHEMATICS 


The  Trigonometric  Functions  of    (J^a)  in  Terms 

of  Functions  of  a 

60.  The  relations  derived  from  the  graphs  of  the 
functions.    Let  the  curves  in  Fig.  45  represent  the  sine 


li- 

f 

-4* 

SB' 

l^ 

.' 

.1 

j? 

JT 

„ 

**■ 

yC 

i  v 

1 

! 

i 

1 

' 

77 

V 

/ 

ir 

V 

A 

4" 

7T 

\ 

V 

yd 

\ 

v 

, 

2 

/ 

/ 

% 

/ 

,lif 

/ 

r 

S 

/ 

/ 

t 

\r 

f 

/ 

s 

4 

/ 

'  f\ 

Jl 

4 

/ 

- 

^ 

"' 

1 lk 

■»» 

^ 

' 

Fig.  45 


and  cosine  functions  drawn  to  the  same  scale,  and  let 

OA±=a  and  OA'  =  90°+a  =  |+a,  a  being  less  than  90°. 

_. 
By  moving  either  curve  a  distance  equal  to  „,  it  can  be 

2» 

made  to  coincide  with  the  other.     Thus  AB  will  coincide 
with  A'B\ 

Hence,  sin  (90°+a)  =  cos  a.  (1) 

Similarly,  the  sine  curve  can  be  made  to  coincide  with 
the  cosine  curve  by  moving  it  to  the  right  a  distance  equal 

IT 

to  g  and  then  rotating  it  about  OX  as  an  axis. 

AC  will  then  coincide  with  A'C 

Hence,       cos  (90°+ a)  =  -  sin  a  (2) 

.'.tan  (90°+ a)  =  -  cot  a        Why  ?  (3) 

and  cot  (90°+a)  =  -tan  a       Why ?  (4) 

Show  that  sec  (90°+ a)  =  -esc  a  (5) 

and  esc  (90°+ a)  =  sec  a  (6) 

Since  A'B'  =  A "£",  it  follows  that 

sin  (90° -a)  =sin  (90°+a)  =  cos  a  (7) 


TRIGONOMETRIC  FUNCTIONS 


47 


Since  A"C"  =  A'C",  it  follows  that 

cos  (90° -a)  =  -cos  (90°+a)  =  +sin  a    (8) 

.*.  tan  (90°  -  a)  =  cot  a        Why  ?  (9) 

and  cot  (90°  -  a)  =  tan  a        Why  ?  (10) 

Show  that  sec  (90°-  o)  =  esc  a  (11) 

and  esc  (90°  —  a)  =  sec  a  (12) 

Sketch  roughly  the  graphs  of  the  tangent  and  cotan- 
gent functions  and  verify  relations  (3)  and  (4). 

The  relations  (1)  to  (12)  may  be  summarized  as  fol- 
lows :  any  trigonometric  functions  of  (90°  —  a)  is  equal  to 
the  cof unction  of  a;'  and  any  trigonometric  function  of 
(90° -\- a)  is  equal  numerically  to  the  cof  unction  of  a,  but 
differs  in  sign  with  the  exception  of  the  sine  and  cosecant 
functions. 

61.  The  relations  found  in  §60  may  be  proved  as 
follows : 

Denoting  ZXAB,  Fig.  46,  by  a,  then  ZABC=^-a. 


a      .  tv 

.  -  =  sm  a  =  cos  \~—a 
c  \2 


b  .     (t 

-  =  cos  a  =  sin     — — a 
c  \2 


r  =  tana  =  cot  (  =  —  a 

-  =  cot  a  =  tan    — — a 
a  \2 

c  /it 

7  =  sec  a  =  csc  (^—  a 

C  fir 

-  =  csc  a  =  sec    -—  a 
a  \2 


Fig.  46 


48 


THIRD-YEAR  MATHEMATICS 


Let    BAD  =  ^,    Fig.    47,    then 
ZXAD=(^+ay 

Prove  that  ACAB^  AEDA. 

.  .  sin  (  o  +  a)  =-  =  +cos  a. 

j         /*  t    \     ~a        a 

ana  cos    77+ a  J  * =  — =  —  sin  a 

\2       /       c  c 


Fig.  47 


EXERCISES 

1.  Prove  relations  (3)  to  (6),  §  60. 

2.  Find  the  exact  values  of  all  functions  of  the  following 
angles:   120°,  135°,  150°. 

3.  Prove  the  relations  sin  (~+ol)  =cos  a 
and  cos  ( J+«)  =  —  sin  a  for  the  following  cases: 

1.  When  a  lies  in  the  second  quadrant,  Fig.  48. 

2.  When  a  lies  in  the  third  quadrant,  Fig.  49. 

3.  When  a  lies  in  the  fourth  quadrant,  Fig.  50. 


Fig.  48 


Fig.  49 


Fig.  50 


TRIGONOMETRIC  FUNCTIONS  49 

The  Trigonometric  Functions  of  In  •  _~  =*=<x  j  in 

Terms  of  the  Functions  of  a 

62.  In  the  exercises  below  it  is  shown  that  we  may 

express  any  function  of  (  n  •  — ±  a  J  in  terms  of  functions  of 

a  by  applying  successively  the  principles  of  §§60  and  61. 
The  results  thus  obtained  are  of  importance  in  making 
trigonometric  tables.  Since  any  angle  greater  than  45° 
may  be  changed  to  the  form  (n-^^a),  a  being  less  than 
45°,  the  functions  of  all  angles  are  expressible  as  functions 
of  a  positive  angle  less  than  45°  and  may  be  found  from  a 
table  giving  only  the  functions  of  angles  from  0°  to  45° > 

EXERCISES 

Give  reasons  for  the  following: 

1.  1.  sin  (180°-a)=sin  [90°+ (90° -a)]  =  cos  (90°-a)  =  sina 

2.  cos  (180°-a)  =  cos[90°+(90°-a)]=  -sin(90°-a)  =  -cosa 

3.  tan  (180°-a)  =  -tana  5.  sec  (180°-a)  =  -sec  a 

4.  COt  (180°-a)  =  -COta  6.   CSC    (180°-a)=CSCa 

2.  1.  sin  (180°+a)  =  -sina  4.  cot  (180°+a)  =  cot  a 

2.  cos  (180°+a)  =  -cos  a  5.  sec  (180°+a)  =  -sec  a 

3.  tan  (180°+a)  =  tan  a  6.  esc  (180°+a)  =  -esc  a 

3.  1.  sin  (270°-a)  =  sin[90o+(180o-a)]  =  cos(180°-a)  = -cosa 

2.  cos  (270° -a)  =  -sin  a  5.  sec  (270° -a)  =  -esc  a 

3.  tan  (270°-a)=cota  6.  esc  (270°-a)  =  -sec  a 

4.  cot  (270°-a)=tana 

4.  1.  sin  (270°+a)  =  -cosa  4.  cot  (270°+a)  = -tan  a 

2.  cos  (270°+a)  =  sina  5.  sec  (270°+a)  =  csc  a 

3.  tan  (270°+a)  =  -cot  a  6.  esc  (270°+a)  =  -sec  a 

5.  1.  sin  (360°-a)=*-sina  4.  cot  (360°-a)  =  -cot  a 

2.  cos  (360°-a)  =  cosa  5.  sec  (360°-a)=sec  a 

3.  tan  (360°-a)  =  -tan  a  6.  esc  (360°-a)  =  -esc  a 


50  THIRD-YEAR  MATHEMATICS 

6.  1.  sin  (360°+a)  =  sina  4.  cot  (360°+a)  =  cot  a 

2.  cos  (360°+a)  =  cosa  5.  sec  (360°+a)  =  sec  a 

3.  tan  (360°+a)  =  tana  6.  esc  (360°+a)  =  csc  a 

63.  From  a  study  of  the  preceding  exercises  we  learn 
the  following : 

1.*  A  function  of  (an  even  multiple  of  90°^  a)  is  equal 
numerically  to  the  same  function  of  a. 

2.  A  function  of  (an  odd  multiple  of  90°  =*=  a)  is  equal 
numerically  to  the  corresponding  cof unction  of  a. 

3.  The  sign  of  the  result  is  the  same  as  the  sign  of  the 
original  function  in  the  quadrant  in  which  the  angle 
(n-£0°±a)  lies. 

EXERCISES 

Express  the  following  as  functions  of  positive  acute 
angles : 

1.  sin  580° 

sin  580°  =  sin  (6  •  90° +40°)  =  -sin  40° 

2.  cos  315°  6.  sin  240° 

3.  sin  (-196°)  7.  tan  (-410°) 

4.  tan  (2tVtt)  8.  cos  120° 

5.  cos  120°  9.  sin  300° 

Express  the  following  as  functions  of  positive  angles  less 
than  45°: 

10.  cos  (-428°)  13.  tan  (-65°) 

11.  sin  (-84°)  14.  sin  1,420° 

12.  esc  (834°)  15.  cot  l,330c 


|0 


*  These  principles  hold  even  when  a  is  not  an  acute  angle.  In 
case  a  is  greater  than  90°  the  sign  is  determined  by  the  quadrant  in 
which  the  angle  (n  •  ^=*=a  j  would  lie  if  a  were  acute. 


TRIGONOMETRIC  FUNCTIONS  51 

Find  the  values  of  the  following: 

16.  cosl80o-sec245o-4sin30o+T/2sin45o4-cosl80ocsc90° 

«_  IT  2tT  .       7T      •       2lT 

17.  cos  -  cos  -——sin  -  sin  — - 

oo  6  6 

,,  —  7T  07T   i  7T      •       07T 

18.  sin  -  cos  — -fcos  -  sin  — 

o  6  bo 

Simplify — 

19.  tan («■■-*);  cos(^-x);   sin  (270°-x);  -cot(90+z); 
cos  (-1,230°) 

Summary 

64.  The  student  should  know  the  meaning  of  each  of 
the  following  terms : 

initial  side  sine  cosecant 

terminal  side  cosine  inverse  trigonomet- 

quadrant  tangent  ric  functions 

trigonometric  cotangent  circular  measure 

functions  secant  radian 

65.  The  following  exercises  review  the  main  topics 
taught  in  the  chapter: 

1.  Give  the  signs  of  the  trigonometric  functions  in  each  of 
the  four  quadrants. 

2.  Show  how  to  find  the  values  of  the  trigonometric  functions 
of  given  angles  by  means  of  a  drawing. 

3.  Show  how  to  construct  an  angle  when  the  value  of  one 
of  the  trigonometric  functions  of  the  angle  is  known. 

4.  Explain  how  the  values  of  the  trigonometric  functions 
of  an  angle  may  be  found  when  the  value  of  one  of  the  functions 
is  given. 

6.  Discuss  the  changes  of  the  trigonometric  functions  as  the 
angle  changes  from  0  to  360°,  using  the  straight-line  representa- 
tion. 


52  THIRD-YEAR  MATHEMATICS 

6.  Give  the  exact  values  of  the  sine  and  cosine  of  the  follow- 
ing angles:  0°,  30°,  45°,  60°,  90°,  120°,  135°,  150°,  180°,  210°, 
225°,  240°,  270°,  300°,  315°,  330°,  360°. 

7.  Prove  the  relations  between  circular  measure  and  degree 
measure. 

8.  Show  that  the  length  of  an  arc  of  a  circle  is  given  by  the 
equation  s  =  rO,  6  being  the  number  of  radians  in  the  angle. 

9.  Draw  the  graph  of  each  of  the  trigonometric  functions 
and  tell  what  the  graphs  show. 

10.  Express  the  functions  of  (—a)  in  terms  of  functions  of  a. 

11.  Express  the  functions  of  (Z^0-)  in  terms  of  functions  of  a. 

12.  Show  how  to  express  the  functions  of  any  angle  as  func- 
tions of  a  positive  angle  less  than  45°. 


CHAPTER  III 

LINEAR  EQUATIONS 

Linear  Equations  in  One  Unknown 

66.  Normal  form.  We  have  seen  that  every  linear 
function  of  x  may  be  changed  to  the  form  ax-\- b,  §  7.  Simi- 
larly every  linear  equation  in  one  unknown  and  many 
fractional  equations  may  be  changed  to  the  form  ax-\-b  =  0. 
This  is  called  the  normal  form  of  a  linear  equation  in  one 
unknown. 

EXERCISES 

Change  the  following  equations  to  the  normal  form  ax-\-b  =  0: 

Subtracting  as  indicated,  — x-\ — - —  =  7. 

Multiplying  by  36,  63z+81-36z+8x-4  =  252. 

Simplifying,  we  have  the  normal  form,  35a:  — 175=0. 


»-I(3«-iHHi-*)=i 


3.  y-\Sy-[±y-(Sy-l)]\  =  l 

4.  (2-2/)(3-2/)  =  (l  — 2/)(5-2/) 

$5.  (2+y)(2y+l)  +  (2-y)(2y+l)-7y  =  0* 
6.  (82/+3)2-32/(132/+9)-(52/+2)2  =  0 

13      3        65  _a_       6       0 

7*  2-5"h4~2z-10  x-6    x-a 

8.  ax+&3  =  6:r+a3  t11-  (a+6)2/  =  2a- (0-6)3/ 

_j, 3^_       _  12'  (^-a)2-(^-6)2  =  c2 

*9'  ^+l~H:2+2~0  13.  (x-a)(x-b)  =  (x-c)(x-d) 

*  All  problems  and  exercises  marked  %  may  be  omitted. 
53 


54  THIRD-YEAR  MATHEMATICS 

67.  Solution  of  linear  equations.  Subtracting  b  from 
both  sides  of  the  equation  ax+b  =  Q  and  then  dividing  by 
a,  we  have  _^ 

jc= — 

a 

This  may  be  stated  in  words  as  follows : 
The  root  of  a  linear  equation  in  normal  form  is  ob- 
tained by  dividing  the  negative  of  the  constant  term  by  the 
coefficient  of  the  unknown  number. 

EXERCISES 

Solve  the  following  equations: 

1.  5l(|-3)  +  .17(2/-2)=3.% 

2.  9.9+7-2y~5-5  =  3.3y-.li/ 

,  2(^)+^->M) 
9  9  5  5 


5. 


x-7^ x-2~ x-S~ x+1 

5  2  2 

H-2    4— 2/2~2/— 2 

2      •  2 

Change  -■; — -;  to  +• 


4_^2  ^2_4 

2x— 2m    n(m—ri)x—m2  ,     m       n 
6.  2         ^       t,      +  — o 

m— n  w?—n2         m—n 

+7      3    _2+l_    22 

+  '*    2+1      2-1       1-22 

2(y-7)        2-y    y+3 

2/2+32/-28i"4-2/    </+7     ' 

32+2    2(2-5)     2-3 
■f9'    22+2  +   1-22    ~z^-z 


10. 


2?/+7_     15        3y-5 
6y+4    8-18?/2    9y-6 


LINEAR  EQUATIONS  55 


til- 


2y-l_     8      _2H-1 
2y+l     l-4i/2    2y-l 


12    5a-3?/_4_y-7a 


Jl3. 


a-36  3a-6 

ra+n  ,      1        m— n 


ra+w       x       m—n 


n2_3n-3m_l     m2_l 
^    '  my         y         m~ny    n 


Problems  Leading  to  Linear  Equations  in  One  Unknown 

68.  Solve  the  following  problems: 

1.  Divide  60  into  two  parts  such  that  the  greater  part 
divided  by  the  smaller  gives  a  quotient  equal  to  2,  leaving  a 
remainder  equal  to  3. 

2.  What  number  must  be  subtracted  from  each  term  of  the 
fraction  ^  in  order  that  the  resulting  fraction  be  equal  to  J  ? 

3.  The  ratio  of  two  numbers  is  a/b.  If  the  first  is  increased 
by  m  and  the  second  decreased  by  n,  the  ratio  of  the  results  is  f . 
What  are  the  numbers  ? 

4.  What  number  must  be  added  to  each  of  the  numbers  7, 
15,  12,  and  24  in  order  that  the  resulting  sums  may  form  a 
proportion  ? 

5.  A  can  do  a  piece  of  work  in  4  days  and  B  can  do  it  in  5 
days.    How  long  will  it  take  both  together  to  do  the  work  ? 

6.  A  can  build  a  fence  in  16  days,  B  in  18  days,  and  C  in  15 
days.    How  long  will  it  require  all  working  together  to  build  it  ? 

$7.  A  can  do  a  piece  of  work  in  50  days  and  B  in  30  days. 
After  working  together  6  days,  A  finishes  the  work  alone.  How 
many  days  does  A  work  alone  ? 

J8.  Three  men,  A,  B,  and  C,  can  do  a  piece  of  work  in  30 
days.  B  can  do  f  as  much  as  A  and  f  as  much  again  as  C.  Find 
the  time  it  would  take  each  to  do  the  work  alone. 


56  THIRD-YEAR  MATHEMATICS 

9.  Two  pipes  can  fill  a  tank  in  2  and  3  hours  respectively. 
How  long  would  it  take  to  fill  the  tank  if  both  pipes  were  open  ? 

10.  A  train  starts  for  a  distant  station  running  at  a  rate  of 
30  mi.  an  hour.  Twenty-one  minutes  later  another  train  starts 
from  the  same  place,  running  in  the  same  direction  at  the  rate 
of  36  mi.  an  hour.  When  will  the  trains  meet  ?  When  will  they 
be  4  mi.  apart  ? 

11.  A  man  travels  to  a  certain  place  at  the  rate  of  18  mi.  an 
hour.  He  returns  on  a  road  6^  mi.  shorter  and  makes  the  trip 
in  20  min.  less  time,  although  traveling  only  17  mi.  per  hour. 
How  long  is  each  road  ? 

12.  A  man  can  row  in  still  water  at  the  rate  of  8  mi.  an  hour. 
It  takes  him  twice  as  long  to  go  5  mi.  upstream  as  it  does  to 
return.     Find  the  rate  of  the  current. 

13.  It  takes  a  man  26  hr.  to  row  downstream  and  back. 
If  he  can  row  8§  mi.  an  hour  in  still  water  and  if  the  rate  of  the 
current  is  4^-  mi.  an  hour,  how  long  does  it  take  him  to  return  ? 

14.  A  man  earned  $90  by  working  a  certain  number  of  days. 
If  he  had  received  75  cents  less  a  day  he  would  have  had  to  work 
10  days  more  to  earn  the  same  amount.  Find  the  number  of 
days  in  each  case. 

15.  A  boy  bought  a  number  of  pencils  for  a  dollar.  Later 
the  price  was  raised  5  cents  per  dozen  and  he  received  12  less 
for  a  dollar.    What  was  the  price  per  dozen  ? 

16.  A  certain  sum  is  divided  among  three  persons,  such  that 
the  first  receives  $20  more  than  the  second  and  the  third  $20 
less  than  the  second.  The  whole  sum  is  $25  more  than  4  times 
as  much  as  the  third  receives.    How  much  did  each  receive  ? 

17.  Into  what  two  parts  must  $5,330  be  divided  so  that  the 
income  of  one  part  at  5  per  cent  shall  be  twice  as  much  as  the 
income  of  the  other  at  4  per  cent  ? 

18.  The  tire  of  the  fore-wheel  of  a  carriage  is  9  ft.,  that  of 
the  hind-wheel  12  feet.  What  distance  will  the  carriage  have 
passed  over  when  the  fore-wheels  have  made  5  more  revolutions 
than  the  hind-wheels. 


LINEAR  EQUATIONS  57 

19.  A  man  when  asked  to  give  his  age  replied:  "Ten  years 
from  now  I  will  be  twice  as  old  as  I  was  ten  years  ago."  How 
old  was  he  ? 

20.  A  box  of  oranges  was  bought  at  the  rate  of  15  cents  a 
dozen.  Five  doz.  were  given  away  and  the  remainder  sold  at 
the  rate  of  2  for  5  cents,  leaving  a  profit  of  30  cents  on  the  box. 
How  many  were  there  in  the  box  ? 

21.  A  father  engaged  his  son  to  work  20  days  on  the  follow- 
ing conditions:  For  each  day  he  worked  he  was  to  receive  $2, 
and  for  each  day  he  was  idle  he  was  to  forfeit  $1.  At  the  end 
of  20  days  he  received  $34.    How  many  days  was  he  idle  ? 

22.  A  number  of  two  digits  is  2  greater  than  17  times  the 
units  digit.     Find  the  number  if  the  sum  of  the  digits  is  8. 

23.  The  tens  digit  of  a  number  of  two  digits  is  6.  If  the 
order  of  the  digits  be  reversed,  the  resulting  number  is  27  greater 
than  the  original  number.     Find  the  number. 

24.  The  length  of  a  circle  is  2-n-r,  r  being  the  radius  and  tt 
being  3.14,  approximately.  The  ratio  of  two  circles  is  3:4 
and  one  radius  exceeds  the  other  by  3.     Find  the  radius  of  each. 

25.  The  angles  of  a  triangle  are  in  the  ratio  2:3:4.  How 
large  is  each  ? 

26.  A  basket  weighing  56  lb.  hangs  on  a  stick  8  ft.  long  at 
a  point  1  ft.  from  the  middle,  while  it  is  being  carried  by  two 
boys,  one  at  each  end.  The  loads  lifted  by  the  boys  are  in  the 
ratio  5:3.     Find  how  much  each  boy  lifts. 

Use  the  law  that  the  algebraic  sum  of  all  leverages  is  0,  for 
balance. 

27.  A  steel  beam  24  ft.  long  and  weighing  966  lb.  is  being 
moved  by  placing  under  it  an  axle  borne  by  a  pair  of  wheels, 
one  end  being  carried.  If  the  axle  is  2  ft.  from  the  middle  of 
the  beam  what  is  the  weight  at  the  end  which  is  carried. 

The  weight  of  the  beam  may  be  treated  as  a  load  of  966  lb. 
hanging  to  the  bar  at  the  midpoint. 


58  THIRD-YEAR  MATHEMATICS 

28.  How  much  water  must  be  added  to  12  gallons  of  a 
25  per  cent  solution  of  alcohol  and  water  to  reduce  it  to  a  10  per 
cent  solution  ? 

29.  How  much  water  must  be  added  to  a  quart  of  a  20  per 
cent  solution  of  ammonia  to  reduce  it  to  a  10  per  cent  solution  ? 

30.  At  what  time  between  three  and  four  o'clock  are  the 
hands  of  a  clock  together  ? 

$31.  At  what  time  between  eight  and  nine  o'clock  are  the 
hands  opposite  each  other? 

$32.  At  what  time  between  five  and  six  o'clock  are  the  hands 
at  right  angles  ? 

33.  If  the  radius  of  a  circle  is  increased  by  3  in.,  the  area  is 
increased  by  60  square  feet.     Find  the  radius  of  the  first  circle. 

34.  By  increasing  each  side  of  a  square  by  2  in.  the  area  is 
increased  by  16  square  inches.     Find  the  side  of  the  square. 

35.  If  19  lb.  of  gold  and  10  lb.  of  silver  each  lose  1  lb. 
when  weighed  in  water,  how  much  gold  and  how  much  silver  is 
contained  in  a  mass  of  gold  and  silver  that  weighs  80  lb.  in  air 
and  72£  lb.  in  water  ? 

$36.  A  can  do  a  piece  of  work  in  a  days  and  B  in  b  days. 
How  long  will  it  take  them  to  do  it  working  together  ?  Use  the 
result  to  solve  problem  5. 

$37.  If  the  radius  of  a  circle  is  increased  by  a  ft.  the  area  is 
increased  by  b  square  feet.  Find  the  radius  of  the  first  circle. 
Use  the  result  as  a  formula  for  solving  problem  33. 

$38.  What  number  must  be  added  to  each  of  the  numbers 
o,  6,  c,  and  d  in  order  that  the  resulting  sums  may  form  a  propor- 
tion ?    Use  the  result  as  a  formula  to  solve  problem  4. 

9 

39.  Solve  the  equation  F  =  ^C+Z2  for  C. 

40.  Solve  s=2(a+Z)  for  I. 

41.  Solve  s=—r  for  a. 


LINEAR  EQUATIONS  59 

42.  Solve  l(W+w')=l'W'  for  w'. 

43.  Solve  PV=pv(l+^)  fort. 


44.  Solve  ^=j^~  for  V. 
V       b—p 

•    mE     . 
46.  Solve  C="5-; for  r. 

46.  Solve  7=7+7  for/2. 
J    h    h 

69.  Historical  note.  The  theory  of  linear  equations  in  one 
unknown  is  very  old.  In  an  ancient  Egyptian  papyrus,  some- 
times called  the  Reckoning  Book  ofAhmes,  written  about  1700  B.C., 
there  is  a  chapter  of  problems  whose  solutions  are  commonly 
referred  to  as  the  "Hau-computations."  Two  samples  of  these 
problems  are: 

I.  "Heap,  its  2/3,  its  1/2",  its  1/7,  its  whole,  it  makes  33," 
and 

II.  "2/3  added  to,  1/3  subtracted  from,  10  remains." 
The  word  "Hau"   (  =  literally  "heap")   meant  the  "un- 
known," and  the  mode  of  thought  employed  in  the  solutions  was 
that  of  the  equation.     For  example,  problem  I  means — 

~^X~j"^X~\~-a-X  1  X  ==  00, 

and  problem  II  means — 

(*+f*)  -i(aH-3*)  =  10 

Though  it  is  known  that  the  Egyptians  as  early  as  1700  B.C. 
and  perhaps  as  early  as  3400  b.c.  could  solve  these  problems,  all 
that  is  known  of  their  method  is  that  it  was  characterized  by 
their  clumsy  calculatory  processes  with  unit-fractions. 

Whether  in  the  large  amount  of  early  mathematical  knowl- 
edge that  passed  from  Egypt  to  Greece  any  of  this  "Hau- 
computation"  was  included  is  not  known.  A  long  interval  had 
to  elapse  before  our  type  of  problem  involving  linear  equations 
in  one  unknown  acquired  a  form  definite  knowledge  of  which 


60  THIRD-YEAR  MATHEMATICS 

has  survived  to  our  day.  This  was  with  Diophantus  of  Alex- 
andria (third  and  fourth  centuries  a.d.).  Certain  references  by 
earlier  writers  lead  us  to  think  that  Diophantus'  method  was 
not  original,  but  was  only  the  culmination  of  methods  that  had 
very  gradually  developed.  Diophantus'  method  is  so  like  ours 
that  we  give  his  rule  in  his  own  words: 

"When  an  equation  is  met  which  contains  the  same  power 
of  the  unknown  on  both  sides  with  different  coefficients,  like 
must  be  subtracted  from  like  till  a  term  becomes  equal  to  a  term. 
But  if  upon  one  or  both  sides  of  the  equation  certain  terms  are 
negative,  the  negative  terms  must  be  added  on  both  sides 
(Operation  I)  until  both  sides  contain  only  positive  terms,  and 
then  like  must  be  subtracted  from  like  (Operation  II)  until  but 
one  term  remains  on  each  side  of  the  equation." 

Let  us  illustrate  by  an  example  in  modern  form: 

8x-ll-2x+5=x-4+3x+10 
Operation   I   j      8x+9  =  6x+2i 


Operation  II   j  2x=u 


Most  of  the  problems  of  the  book  that  are  devoted  to  this 
type  lead  to  the  form  axm  =  b,  (m=  1). 

The  Hindus  went  beyond  Diophantus  by  introducing  the 
idea  of  negative  number,  so  that  with  them  Diophantus'  Opera- 
tion I  was  not  needed  nor  used.  The  Hindus'  use  of  the  phrase 
"subtracting  similars"  is  so  like  the  Diophantine  "subtracting 
like  from  like"  that  historians  think  the  Hindus  were  never- 
theless influenced  by  Greek  writings. 

The  Arabs  were  the  next  to  work  on  the  theory  of  equations. 
They  were  strongly  influenced  by  Greek  mathematics  and  were 
never  able  to  rise  to  the  notion  of  purely  negative  numbers. 
Consequently  they  always  prescribed  the  Operations  I  and  II 
of  Diophantus.  The  Arabs  called  one  of  these  operations 
aldschebr,  whence  our  word  algebra,  meaning  restoring,  and 
the  other  mukdbala,  meaning  opposing.  We  say  transposing  and 
combining.    Algebra  took  its  name  from  this  phrase.    Modern. 


LINEAR  EQUATIONS  61 

knowledge  starts  with  Arabic  sources  (Tropfke,  Geschichte  der 
Elementar-Mathematik,  Band  I,  pp.  242-47.) 

Linear  Equations  in  Two  Unknowns 

70.  Graph  of  a  linear  equation  in  two  unknowns.    In 

an  equation  containing  two  unknowns,   as  2x-\-4y  =  l, 

either  unknown  may  be  regarded  as  a  function  of  the  other. 

„.  \-2x         l-4i/ 

Thus,  y  =  -^-,x  =  -^-. 

In  general,  if  ax-\-by  =  c,  then  y  =  — =- — ,  x  — -. 

O  (X 

Show  that  y  is  a  linear  function  of  x  and  that  x  is  a  linear 
function  of  y. 

We  have  seen,  §  8,  that  a  linear  function  is  represented 
graphically  by  a  straight  line.     The  straight  line  repre- 

senting  y  =  —7 —  is  also  said  to  be  the  graph  of  the  equa- 
tion ax+by  =  c. 

EXERCISES 

Graph  the  following  equations: 

1.  2x+y  =  5 

Let  x  =  0,  then  y  =  5;  let  y  =  0,  then  z  =  2.5.  Plot  the  points 
determined  by  these  pairs  of  numbers  and  draw  the  straight  line 
passing  through  them. 

2.  3x+7?/  =  42  4.  5z  =  27-2?/ 

3.  x-\8=-3y  6.  4-3x-?/=0 

71.  Graphical  solution  of  a  system  of  equations.    A 

set  of  values  of  the  unknowns  which  satisfies  both  equations 
is  a  solution  of  the  system.  Since  the  graph  of  a  linear 
equation  in  two  unknowns  is  a  straight  line  and  since 
two  straight  lines  either  are  different  and  have  at  most 
one  point  in  common,  or  are  the  same  and  have  all  points 
in  common,  a  system  of  two  linear  equations  cannot  have 


62 


THIRD-YEAR  MATHEMATICS 


more  than  one  solution.  The  co-ordinates  of  the  point 
of  intersection  of  the  two  straight  lines  are  the  solution 
of  the  system. 

EXERCISES 

Solve  the  following  systems  graphically: 
fx+y=5 


(x,  y)  =  (4,  1)  is  the  solution  of  the  system 
(x+y  =  4 

\y-x=2 


x-y  =  3 
In  x+y  =  5,  let  x  =  0,  then  y  =  5;  let 
y  =  0,  then  x  =  5 .  These  pairs  of  values  of  x 
and  y  determine  two  points.  The  straight 
line,  AB,  Fig.  51,  passing  through  these 
points  represents  the  equation  x+y  =  5. 
Similarly  draw  CD,  the  graph  of  x  —  y  =  Z. 
The  co-ordinates,  4  and  1,  of  the  point  of 
intersection,  P,  are  the  required  solution, 
.e.,  (x,  y)  =  (4,  1)  is  the  solution  of  th«  svstpm  j*+2/  =  5. 

\x-y  =  3 

(3x+  y  =  9 
[5x-3y=l 

(3H-s=l'4  r&r  -by  =14 

*  \2y-5x=-19  5*   \7x+2y  =  32 

72.  Solution  of  a  system  of  linear  equations  by 
determinants.  In  a  given  system  of  linear  equations  in 
two  unknowns  the  terms  containing  the  unknowns  may 
be  brought  to  one  side,  and  the  terms  not  containing  the 
unknowns  to  the  other  side,  of  the  equation.  After  all 
similar  terms  have  been  combined,  the  system  is  of  the 
following  normal  form: 

f  ax+  by  =  c 

\aix+biy  =  ci 

To  eliminate  y,  the  first  equation  is  multiplied  by  fci 
and  the  second  by  b.    This  gives  the  equations, 
(abix+bbiy  =  cbi 
\aibx+biby  =  cib 


GUILLAUME  FRANCOIS  ANTOINE  L'HOPITAL 


Guillaume    Francois    Antoine    THopital 

^lUILLAUME  FRANCOIS  ANTOINE 
^J"  L'HOPITAL  was  born  at  Paris  in  1661  and 
died  there  in  1704.  He  was  one  of  the  pupils 
of  John  Bernoulli,  through  whom  he  became  one 
of  the  earliest  appreciators  of  the  infinitesimal 
calculus,  then  a  science  so  new  that  no  texts  had 
been  written  upon  it. 

He  wrote  the  first  treatise  on  the  new  science 
in  1696  under  the  title  Analyse  des  infiniment 
petits.  The  wide  circulation  of  this  book  brought 
the  differential  notation  of  Leibnitz  into  general 
use  in  France  and  helped  to  make  it  known  in 
Europe.  He  took  part  in  many  of  the  challenges 
of  the  friends  of  Leibnitz  and  Newton. 

L'Hopital  also  wrote  a  treatise  on  conic  sec- 
tions, which  was  not  published  until  1707,  three 
years  after  the  author's  death.  In  this  he  treated 
the  sub j  ec t  analytically.  This  means  that  he  used 
algebraic  methods  to  derive  the  properties  of  the 
conic  sections.  He  did  his  work  so  well  that  his 
book  was  regarded  as  a  standard  on  the  subject 
for  nearly  a  century.  He  was  not  a  teacher  by 
profession, 'but  through  his  excellent  texts  and 
treatises  he  became  one  of  the  greatest  teachers 
of  subsequent  times.  Mathematical  study  and 
textbook  writing  were  his  avocation,  and  his 
title  to  fame  rests  on  what  he  achieved  in  this 
avocation. 

[See  Ball,  5th  ed.,  pp.  369-70.] 


LINEAR  EQUATIONS  63 

Subtracting  the  second  equation  from  the  first  and 
dividing  by  the  coefficient  of  x, 

cbi  —  cib 

x  == 

abi  —  aib 

Similarly, 

_  ac\  —  die 

abi  —  aib 

It  will  be  shown  below  how  these  results  may  be  used  as 
formulas  to  find  the  solution  of  the  system  directly  from 
the  given  equations. 

Each  of  the  expressions  cbi  —  c\b,  abi  —  aib,  and  ac\  —  a\C 
is  of  the  form  of  the  difference  of  two  products.  Such 
expressions  are  called  determinants* 

*  Leibnitz  in  a  letter  to  L'Hopital,  of  April  28,  1693,  was  the 
first  to  publish  the  essential  features  of  the  methods  of  solution  of 
equations  by  determinants,  though  his  procedure  was  somewhat 
different  from  the  modern  form.  He  also  drew  attention  to  the 
importance  of  the  theory  of  permutations  and  combinations  in 
determining  the  factors  and  signs  of  the  products.  Beyond  these 
announcements  about  the  method,  Leibnitz  did  nothing  further  with 
it,  nor  did  any  of  his  contemporaries.  Aside  from  a  "Note"  in  a 
mathematical  journal  of  1700,  nothing  further  was  heard  of  the 
method  until  Gabriel  Cramer  in  an  appendix  of  his  book  of  1750  on 
The  Analysis  of  Curves  solved  a  system  of  n  equations  in  n  unknowns 
by  the  method,  showed  how  to  use  the  theory  of  combinations  and 
permutations  with  it,  and  convinced  men  of  its  power. 

Bezout  (1730-83)  and  Vandermonde  (1735-96)  both  worked 
on  the  theory  of  determinants  and  Laplace  made  important  applica- 
tions of  it.  Lagrange  (1736-1813)  applied  the  doctrine  to  the 
problems  of  analytical  geometry  and  Gauss,  in  1801,  made  important 
investigations  and  improvements  in  the  new  theory.  The  moderi? 
name  determinants  is  due  to  Cauchy  (1789-1857).  Jacobi  (1804-51) 
completed  the  theory  of  determinants.  The  classic  texts  on  the 
subject  are  Brioschi's  of  1854,  Baltzer's  of  1857,  Scott's  of  1880, 
and  Muir's  of  1882.     (Tropfke,,  I,  143-46.) 


64 


THIRD-YEAR  MATHEMATICS 


The  determinant  cbi  —  cib  may  be  represented  by  the 
following  symbol : 

c    b 

Ci     61 

which  means  that  from  the  product  cbi  we  are  to  subtract 
the  product  bci. 

Similarly  ahi—aib  and  oci— aiC  may  be  written 


a    b 
ai   bi 

and 

a    c 
ai  ci 

Hence  the  solution  of  the  system 

(ax+by  =  c 
\aix+biy  =  a 

takes  the  form 

c     b 

a    c 

•T»  — 

Ci     61 

ai   Ci 

•L  — 

a    b 

f 

y— 

1  a    b 

ai   I 

\ 

1  ai   bi 

Notice  that  the  two  denominators  are  the  same,  the 
numbers  in  the  first  column  being  the  coefficients  of  x 
and  the  numbers  in  the  second  column  the  coefficients 
of  y  in  the  given  equations.  This  makes  it  easy  to  remem- 
ber the  denominator.  The  numerator  of  the  fraction 
which  gives  the  value  of  x  is  obtained  from  the  denomi- 
nator by  replacing  the  numbers  in  the  first  column  (the 
coefficients  of  x)  by  the  constants  c  and  ci  respectively. 
The  numerator  of  the  fraction  which  gives  the  value  of 
y  is  obtained  from  the  denominator  by  replacing  the 
numbers  in  the  second  column  (the  coefficients  of  y)  by 
c  and  ci. 


LINEAR  EQUATIONS 


65 


EXERCISES 

Solve  the  following  systems: 

(4x+6y  =  9 
X*  \2x+9y  =  7 
19    61 
7    9 1    9.9-6-7    81-42 


14    6 

4-9-6.2    36-12 

\2    9 

14    9 

\2    7 

4-7-9-2     10     5 

39  =  13 

:24     8 


24 


13      5 


4    6 
2    9 

Hence,  fei/)  =  (|,  ^) 

(2x+Sy  =  Q 
A'  \3x-5?/  =  4 

/5z+  ?/=9 
"■  \Sx+  y  =  5 


6. 


24     12 


10. 


f  2z  =  53  +2/ 
\19x-17y  =  0 
(ax—by  =  c 
'  \dx-ey=f 


a    o 

4      6_1 
5*    3y 
6"~4~2 


ax—by  =  0 
x—  y  =  c 


+  *  \3x— 4?/  =  4n 


JH. 


12. 


13. 


14 


(a+b)x+(a—b)y- 
(a—b)x—(a-\-b)y- 
Mx+y)+b(x-y) 
'  \(a+b)x-(a—b)y- 
115    f(a+6)x— (a—6)y: 


{3x-{-4m2/=8mn 


y 

4a& 

2a2-262 
■a 
b 

■2ac 
2ab 


66 


THIRD-YEAR  MATHEMATICS 


73.  Inconsistent  and  equivalent  equations.  We  have 
seen  that  two  linear  equations  in  two  unknowns  are 
represented  graphically  by  two  straight  lines,  and  that 
the  co-ordinates  of  the  point  of  intersection  form  the 
solution  of  the  system.  However,  two  lines  do  not  always 
intersect:  they  may  be  parallel  or  they  may  coincide. 


1.  If  we  graph  the  system 


2x+y  =  5 


Qx+3y=18 
parallel  lines,  Fig.  52.     In  this  case 
the  equations  have  no  common  solu- 
tion.   They    are    said    to   be   incon- 
sistent, or  incompatible. 

Moreover,  if  we  solve  the 
same  system  by  determinants,  we 
have 


we  obtain  two 


X' 


x  — 


5     1 

2      5 

18    3 

-3 

6     18 

6 

2     1 

7  0  *  y       o        o 

6    3 

- 

rr* 

3£ 

S 

j£ 

s 

3 

£ 

$Z 

>S 

5 

\V 

-  ^  ■ 

'       -p- 

-K 

L 

j£ 

5 

^ 

Fig.  52 


Since  it  is  impossible  to  divide  the  numbers  —3  and 
6  by  zero  the  resulting  forms  show  that  the  equations 
have  no  common  solution. 


graph    the    system 
we  find  that  both   are 


2.  If    we 
/  *-  y=  2 
\5x-5y=l0 

represented  by  the  same  line,  Fig.  53. 
Henoe  any  solution  of  either  equation 
is  a  solution  of  the  other.  Such 
equations  are  said  to  be  equivalent 
or  dependent. 

The  difficulty  that  arises  here  is  not  that  there  is  no 
solution,  but  that  there  are  too  many. 


51       -7 

7 

2 

7 

~7 

xj.   -f>    Z          x 

2 

/ 

£     1 

Fig.  53 


LINEAR  EQUATIONS 


67 


Solving  the  same  system  by  determinants,  we  have 


2 

-1 

10 

-5 

1 

-1 

5 

-5 

-10+10    0 


2 
10 


-5  + 


5  =  0'  V  = 


Since  a  number  multiplied  by  zero  always  gives  zero, 

the  expression  ^  may  represent  any  number.     Hence  the 

solution  is  indeterminate.  It  is  easily  seen  that  one  equa- 
tion may  be  derived  from  the  other  by  simple  multiplica- 
tion by  a  constant. 

The  two  preceding  examples  show  that  a  system  of  the 
form 

j  ax-\-  by  =  c 

\aix+biy  =  ci 

has  one,  and  only  one,  solution,  if  the  determinant 
abi  —  aib  is  not  equal  to  zero.  Hence  this  fact  may  be  used 
to  determine  whether  a  system  of  equations  has  one  and 
only  one  common  solution. 

EXERCISES 

Show  which  of  the  following  systems  are  equivalent,  and 
which  inconsistent. 


1. 


2.  { 


3x+|=6 

'3x-2y=U 
<9x-Qy  =  3G 

2 

x+jV  =  2 

2+7</=l 


4. 


7. 


(3x+2y-7-x=  12-3?/ 
\2x+5?/  =  20 

(7x-8  =  4y-2x 
\18x-8y=W 

(3x+±y=12 
\6x+8?/=14 

(x-y+l  =  0 


68  THIRD-YEAR  MATHEMATICS 

Linear  Equations  with  Three  or  More  Unknowns 

74.  Solution  by  elimination.  Systems  of  equations  in 
three  or  more  unknowns  are  solved  by  the  methods  used 
in  solving  equations  in  two  unknown  numbers.  In 
general,  the  aim  should  be  to  obtain  first  two  equations 
in  two  unknowns  by  eliminating  the  third  unknown,  and 
then  to  solve  these  two  equations. 

EXERCISES 

Solve  the  following  systems: 

Ux-  y+  z=l 

1.  \  x+2y+7z  =  7 
[Sx  —  y  —  5z  =  5 

Subtracting  the  third  equation  from  the  first, 
£+62=  —4 

Multiplying  the  third  equation  by  2  and  adding  the  resulting 
equation  to  the  second  equation, 

7x-Sz  =  17 

Solving  the  system 

fz+63=-4 
\7x-3z  =  17 
we  have  (x,  z)  =  (2,  —1). 

By  substituting  these  values  in  the  first  equation,  we  find 
2/  =  6. 

.*.  {x,  y,  z)  =  (2,  6,  —1)  is  the  solution  of  the  system. 

(2a-b+c=l  (x+2y-z=2 

2.  \a-7b-8c=l  4.  <3x-2y+2z  =  0 


l7a+146+2c  =  7  [5x-4:y+3z=l 

(x+2y-4z=ll  (5x-7y-z=  16 

<2x  =  Sy  6.  hx-2y+2z=U 

[y-4z=0  [2x+y+3z  =  6 


LINEAR  EQUATIONS 


69 


75.  Determinant  of  the  third  order.    The  symbol 


«1 

61 

Ci 

a2 

&2 

C2 

«3 

63 

C3 

is  called  a  determinant  of  the  third  order.  It  represents 
the  following  sum: 

G1&2C3  +  (hhCi  +  (J3&1C2  —  Ci&2tt3  ~~  C2^3«l ""  C302&1- 

The  nine  numbers  ah  02,  a>z,  h,  b2)  63,  etc.,  are  called  the 
elements.  The  horizontal  lines  in  the  square  form  are  the 
rows  and  the  vertical  lines  the  columns  of  the  determinant. 
Each  term  in  the  expansion  is  a  product  of  three  ele- 
ments, no  two  of  which  lie  in  the  same  row  or  in  the 
same  column. 

A  determinant  of  the  third  order  may  be  expanded  as 
follows : 

Draw  the  diagonal  through  the 
first  element,  ai,  Fig.  54,  and  the 
parallels  to  it  through  02  and  a3  respec- 
tively. This  gives  the  terms  ai&2C3, 
a2&3Ci,  and  a^h. 

Then  draw  the  diagonal  through 
C\  and  the  parallels  through  c%  and  c3.   - 
The  signs  of  the  last  three  products  are  changed, 
gives  the  terms  —  cib2ad,  —  c^hai,  and  —CzChbi. 


Fig.  54 


This 


EXERCISES 

Evaluate  the  following  determinants: 

5        2-6 

1  4        7  =5-4-  1+1  -3-  (-6R2-7-2 

2  3        1  -(-6) -4.2-7- 3- 5-1- 1-2 
=  20-18+28+48-105-2 
=  -29 


70 


THIRD-YEAR  MATHEMATICS 


2. 


1        3        8 

1 

1 

2 

-12         0 

4. 

1 

1 

8 

1     -4         5 

1 

-1 

0 

1         1    .     11 

1 

2 

1 

1     -1     -1 

5. 

3 

7 

3 

8        3        0 

4 

3 

5 

76.  Solution  by  determinants.    By  solving  the  equa- 
tions 

(aix+biy+ciz  =  di 
la2X-\-b2y+C2Z  =  (k 
{azx+hy+czz  =  dz 

a  formula  may  be  obtained  for  the  solution  of  any  system 
of  three  linear  equations  in  three  unknowns. 

Eliminating  y  between  the  first  two  equations,  we  have 


{aih.  —  avbi)x+  (b2Ci  —  6102)2/  =  di&2 _  #261- 


(i) 


Eliminating  y  between  the  first  and  third  equations, 
we  have 


(o36i — aib3)x+  (c36i  —  hd)y  =  d36i  —  dib3. 
Solving  equations  (1)  and  (2),  we  have 

d\b2Cz + a\bzd + dzc^bi  —  C\b2dz  —  C2&3CJ1  —  Cza\b\ 


(2) 


x  = 


O162C3 +0263(4  +O3C261  —  cib2a3  —  C2&3O1  —  c3O20i 
According  to  §  75  this  may  be  written 


di 

61 

Ci 

k 

b2 

C2 

Us 

63 

Cz 

ai 

61 

Ci 

02 

62 

Ol 

o3 

63 

Cz 

LINEAR  EQUATIONS 


71 


Notice  that  the  denominator  is  a  determinant  whose 
elements  are  the  coefficients  of  x,  y,  and  z  in  the  given 
system  and  that  the  numerator  is  derived  from  the  de- 
nominator by  replacing  the  coefficients  of  x  by  the 
constants. 

Similarly, 


ai    di 

Ci 

ai 

h    di 

02    a\ 

C2 

a* 

62    a\ 

a%    d% 

Cz 

.,  z  = 

az 

bz    dz 

ai    bi 

Ci 

ai 

bi     ci 

02       62 

Ol 

«2 

62      Ol 

a$    63 

c3 

0*3 

bz     cz 

EXERCISES 


Solve  by  determinants: 

[2z+32/+4z=16 
1.  <5x-8y+2z  =  l 
{3x-y-2z=5 


16 

3 

4 

1 

-8 

2 

5 

-1 

-2 

2 

3 

4 

j  y—~ 

5 

-8 

2 

3 

-1 

-2 

2  16        4 
5        12 

3  5-2 


2  3        4 

5-8        2 

3  -1     -2 

.-.(*,  y,  z)  =  (3,  2,1) 


2 

3 

16 

5 

-8 

1 

3 

-1 

5 

2 

3 

4 

5 

-8 

2 

3 

-1 

-2 

(5x+2y-4:Z=-3 
2.  !4x+5i/+2z  =  20 
l3x-3!/+5z=12 


3. 


Sx-y+2z=9 
z-2?/+3z=2 
2x-32/+3=l 


fa+36+9c  =  23 
4.  <L+26+4c=15 
L+6+c  =  9 

6.  |a+36  =  4 
U-2c  =  6 


72 


THIRD-YEAR  MATHEMATICS 


PROBLEMS   AND   EXERCISES 

77.  Solve  the  following : 
7x+8    7y-l 


2s-4    y-1 

o      "T     o 


=  -2 

1 

3 


2. 


J3. 


2/y 


4a— 2y 
23-2/ 


=  2y-19 


f2+x     2-y_3(y-4aQ 


J5. 


*6. 


1 


4  5 


x-y    x+y 
x    y 


=  15 
=  17 


3?   y 


6 


3 
z-3 


5=^-3(y-z) 


[V  =  12 

x    y 


2    3 


[2x+3?/+5  =  0 

7.  J6?/+52  =  7 
l3z+10z=l 
fx+2?/+z=-17 

8.  <2x+y-z=-l 
{3x-y+2z  =  2 

3  5 


=  14 


9. 


Do  not  clear  of  fractions. 

Regard  -  and  -  as  the  unknowns. 
x         y 


4x  —  y    2x  —  y 
3  4        23 


y—2x    y—Ax     5 


10.  A  mixture  of  alcohol  and  water  contains  10  gallons. 
A  certain  amount  of  water  is  added  and  the  alcohol  is  then 
30  per  cent  of  the  total.  Had  double  the  amount  of  water  been 
added  the  alcohol  would  then  have  been  20  per  cent  of  the  whole. 
How  much  water  was  actually  added  and  how  much  alcohol 
was  there  ?     (Board.) 

11.  The  value  of  146  francs  is  as  great  as  that  of  117  shillings. 
A  dollar  and  4  francs  together  are  worth  32  cents  more  than  6 
shillings.  Find  the  value  in  cents  of  a  franc  and  a  shilling. 
(Board.) 

12.  A  photographer  has  two  bottles  of  diluted  developer. 
In  one  bottle  10  per  cent  of  the  contents  is  developer  and  the 


LINEAR  EQUATIONS  73 

rest  water;  in  the  other  the  mixture  is  half  and  half.  How 
much  must  he  draw  from  each  bottle  to  make  8  oz.  of  a  mixture 
in  which  25  per  cent  is  developer  ?     (Board.) 

13.  A  principal  of  $2,500  put  at  simple  interest  and  for  a 
certain  time  amounts  to  $2,800.  If  the  rate  of  interest  had  been 
1  per  cent  higher  and  the  time  two  years  longer,  the  amount 
would  have  been  $3,200.     Required  the  time  and  rate.     (Board.) 

14.  A  certain  number  of  bolts  can  be  bought  for  a  dollar. 
If  10  more  could  be  bought  for  a  dollar  the  price  would  be 
half  a  cent  less  per  dozen.  What  is  the  price  per  dozen? 
(Board.) 

15.  A  man  travels  50  mi.  in  an  automobile  in  Sj  hours.  If 
he  runs  at  the  rate  of  20  mi.  an  hour  in  the  country  and  at  the 
rate  of  8  mi.  an  hour  when  within  city  limits,  how  many  miles 
of  his  journey  is  in  the  country  ?     (Yale.) 

16.  A  company  contracted  to  make  252  automobiles.  Two 
factories,  working  together,  can  make  this  number  in  12  days. 
Working  alone,  one  factory  requires  7  days  longer  than  the  other 
to  do  this  amount.  Find  the  time  in  which  each  factory  alone 
can  fulfil  the  contract.     (Sheffield.) 

17.  A  and  B  together  can  do  a  piece  of  work  in  12  days. 
After  A  has  worked  alone  for  5  days,  B  finishes  the  work 
in  26  days.  In  what  time  can  each  alone  do  the  work? 
(Sheffield.) 

18.  Two  yachts  race  over  a  48-mile  course.  Owing  to  differ- 
ence in  measurement,  B  is  given  a  start  of  half  a  mile  in  the  first 
trial  and  is  beaten  by  6  minutes.  In  the  second  trial,  the  rate 
of  the  wind  being  the  same  as  before,  B's  start  is  increased  to  a 
mile  and  a  half,  and  still  A  wins  by  2  minutes.  Find  the  rate 
in  feet  per  minute  of  each  boat.     (Chicago.) 

19.  A  sum  of  $1,050  is  divided  into  two  parts  and  invested; 
the  simple  interest  on  the  one  part  at  4  per  cent  for  6  yr.  is 
the  same  as  the  simple  interest  on  the  other  at  5  per  cent  for 
12  yr.;  find  how  the  money  is  divided.     (Princeton.) 


74  THIRD-YEAR  MATHEMATICS 

20.  A  man  has  two  sons,  one  six  years  older  than  the  other. 
After  two  years  the  father's  age  will  be  twice  the  combined  ages 
of  his  sons,  and  six  years  ago  his  age  was  four  times  their  com- 
bined ages.    How  old  is  each  ?     (Princeton.) 

21.  In  buying  coal  A  gets  1  ton  more  for  $18  than  B  does; 
he  pays  $9  less  for  6  tons  than  B  pays.  Find  the  price  per  ton 
that  each  pays.     (Princeton.) 

22.  In  paying  two  bills  aggregating  $175,  a  merchant  availed 
himself  of  discount  for  cash,  10  per  cent  on  one  and  5  per  cent 
on  the  other,  and  then  paid  them  both  with  $166.  What  was 
the  amount  of  each  bill  ?     (Chicago.) 

23.  Two  locomotives,  A  and  B,  are  on  tracks  which  cross 
each  other  at  right  angles.  When  B  is  at  the  point  of  crossing, 
A  has  675  ft.  yet  to  go  before  reaching  this  point.  In  5  sec. 
the  two  locomotives  are  at  an  equal  distance  from  the  crossing, 
and  in  40  sec.  more  they  are  again  at  an  equal  distance  from  it. 
What  is  the  rate  of  each  in  feet  per  second?  Illustrate  by  a 
diagram. 

24.  A  dealer  has  two  kinds  of  coffee,  worth  30  and  40  cents  per 
pound  respectively.  How  many  pounds  of  each  must  be  taken 
to  make  a  mixture  of  70  lb.  worth  36  cents  per  pound  ?    (Yale.) 

25.  A  man  bought  a  certain  number  of  eggs.  If  he  had 
bought  88  more  for  the  same  money  they  would  have  cost  him 
less  by  a  cent  apiece;  if  he  had  bought  56  fewer  they  would  have 
cost  more  by  a  cent  apiece.  How  many  eggs  did  he  buy  and 
at  what  price  each?     (Yale.) 

26.  An  investment  at  simple  interest  for  6  yr.  amounts  to 
$4,960.  If  the  rate  had  been  1  per  cent  greater  the  amount 
would  have  been  $5,000  in  5  years.  Find  the  rate  and  the  sum 
invested.     (Chicago.) 

27.  The  sum  of  the  three  digits  of  a  number  is  16.  The 
sum  of  the  first  and  third  digits  is  equal  to  the  second;  and  if 
the  digits  in  the  units  and  in  the  tens  places  be  interchanged  the 
resulting  number  will  be  27  less  than  the  original  number. 
What  is  the  original  number  ? 


LINEAR  EQUATIONS  75 

28.  A  chauffeur  engages  to  accomplish  a  journey  of  105  mi. 
in  a  specified  time.  After  traveling  63  mi.  uniformly  at  a  rate 
which  will  just  enable  him  to  keep  his  agreement,  his  car  is 
delayed  24  minutes.  He  then  drives  3^  mi.  faster  per  hour  than 
before  and  arrives  exactly  on  time.  What  was  his  original  rate  ? 
(Board.) 

Summary 

78.  The  chapter  has  taught  the  meaning  of  the  follow- 
ing terms.     Give  the  meaning  of  each: 

normal  form  of  a  linear  determinant  of  the  second 

equation  order,  of  the  third  order 

graph  of  an  equation  inconsistent  equations 

solution   of  a   system   of  equivalent  equations 
equations 

79.  Tell  how  to  solve  a  system  of  linear  equations  in 
two  unknowns  (1)  graphically,  (2)  by  elimination, 
(3)  by  determinants. 

80.  Tell  how  to  solve  a  system  of  linear  equations  in 
three  unknowns  (1)  by  elimination,  (2)  by  determinants. 


CHAPTER  IV 
QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN 

Methods  of  Solving  Quadratic  Equations 

81.  Quadratic  equation.     Equations  like 

x2— 5z  =  6,        %+5x2  =  x,        mx2  —  nx  =  7—Sx, 
x2+ax—abc-\-a(b-\-cx+2x2)=0, 

are  quadratic  equations. 

By  bringing  all  terms  to  one  side  of  the  equation  and 
then  collecting  similar  terms,  show  that  -these  equations 
may  be  changed  respectively  to : 

x2-5z+6  =  0;  5x2-x+%  =  0;  mx2-(n-S)x-7  =  0; 
(l+2a)x2+(a+ac)x+(ab-abc)=0. 

In  general,  any  quadratic  equation  in  one  unknown 
can  be  changed  to  the  following  normal  form: 
ax2+bx+c=0, 

where  a  denotes  the  coefficient  of  the  term  in  x2,  b  the 
coefficient  of  the  term  in  x,  and  c  the  sum  of  the  terms  not 
containing  x,  i.e.,  the  sum  of  the  constant  terms. 

Give  the  values  of  a,  b,  and  c  in  each  of  the  equations  above. 

82.  Methods  of  solving  quadratic  equations.  In 
§§13  and  23  quadratic  equations  were  solved  by  graph 
and  by  factoring. 

The  method  by  factoring  has  the  advantage  of  being 
brief  and  is  used  when  the  quadratic  function  forming  the 
first  member  is  readily  factored. 

We  cannot  solve  every  quadratic  equation  by  the 
graphical  method.     For  example,  let  us  try  to  use  the 
graphical  method  to  solve  the  equation: 
z2-4z+5  =  0. 
76 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN      77 


X 

m 

10 
5 
2 
.1' 
2 
5 

1(T 

I 

•ho 

1 

/<* 

-i 

0 

i 

1 

\ 

2 

\ 

6 

\ 

4 
5 

r, 

; 

v 

/ 

\ 

/ 

*s 

J 

0 

5 

X 

Fig.  55 


Let  /(z)=x2-4x+5. 

The  table,  Fig.  55,  gives  the  values  off(x)  correspond- 
ing to  integral  values  of  x  between 
—  1  and  +5.  It  is  seen  that  the 
graph  of  }{x)  and  the  x  —  axis  have 
no  points  in  common.  Hence  the 
graph  does  not  enable  us  to  find 
the  roots  of  the  equation  x2— 4x 
+5  =  0. 

However,  any  quadratic  equa- 
tion can  be  solved  either  by 
completing  the  square  or  by  the 
formula.    These  methods  are  therefore  general. 

83.  Solution  by  completing  the  square*  and  by 
formula.  Since  every  quadratic  equation  may  be  changed 
to  the  normal  form, 

ax2+bx+c=0> 

we  may  obtain  a  solution  of  every  quadratic  equation  by 
solving 

ax2+bx+c  =  0. 

This  will  lead  to  a  formula  for  finding  the  roots. 

Subtracting  c  from  both  sides  of  the  equation 
ax2+6x+c  =  0,  and  dividing  by  a,  we  have 


X2-\ — £=  — 


b2 


Completing  the  square  by  adding  ^  to  both  sides, 


x2+-x+—t ■■■ 
a      4a2 


4a2' 


*  This  method  originated  with  the  Hindus.  Aryabhatta 
(b.  476  a.d.)  first  used  it  in  a  slightly  different  form  from  that  given 
here.  Brahmagupta  (b.  598  a.d.)  used  it  so  extensively  that  it 
has  been  given  the  name  Brahmagupta's  Rule,  and  Cridhara  later 
modified  it  slightly,  bringing  it  to  the  modern  form.  (Tropfke,  Band 
I,  S.  257.) 


78  THIRD- YEAR  MATHEMATICS 

Extracting  the  square  root, 


,   b  lb2-4ac 

x+2a=  *  visr 

Hence    Xi=~b+V¥^ac f    ft_-»-^Zg 

EXERCISES 

Solve  the  following  equations  by  formula: 

1.  3z2+5x-2  =  0 

Comparing  this  equation  with  the  equation  ax2-\-bx+c-0,  we 
find  that 

o=3;  6=5;  c=  —2 

Substituting  these  values  in  the  formulas, 

-5=  V  25+24     -5=7     1        ,      . 
x- ^ =  — — --,  and  -2 

2.  2^+5a;+2  =  0  10.  r2-3.50r+2.80=0 

3.  2r2— r  —  6  =  0  Give  values  correct  to  three 

4.  lAx2-\-5x  =  2A  significant  figures. 

6.  6p2-13p=10p-21  jn.  t2=  .100-  .200t 
$6.  my2+ny+p  =  0  12>  16. 08^+20^=1,000 

7.  ax2+(b-a)x-b  =  0  13    (m-n)y2-m2y+m2n  =  0 

8.  a-y2=(l-a)y  (2y-l)(y-S)  =  2 

9.  y2-l.Gy+0.3  =  0 

«.         .  ++  15.  (z-l)2(z+3)=z0c+5)(x-2) 

Give  values  correct  to  '  '. 

two  significant  figures.       J16.  ?/2-6mn/+ra2(9r2-4n2)  =  0 

17.  m2y2-  (m2+mn)y  =  2m2  -  5mn+2n2 

The  formula  gives: 

m2+mn=/m4+2m3n+m2n2+8m4-20m3n+8m2n2" 
V  2m* 

_  m2  +mn  =  /9m4  -  18m3n +9m2n2 
2m2 

m2+ran  =  (3m2— 3mn) 

= ~= -,  etc. 

2m2  ' 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN       79 

In  solving  exercise  17  it  is  necessary  to  find  the  square 
root  of  the  polynomial  9m4— 18m3n+9m2n2.  This  is 
readily  done  by  inspection. 

It  is,  however,  not  always  easy  to  find  by  inspection 
the  square  root  of  a  polynomial.  Hence  we  shall  learn 
the  process  of  extracting  the  square  root  of  a  polynomial 
before  proceeding  with  the  general  solution  of  quadratic 
equations. 

Square  Root  of  Polynomials 

84.  The  process  of  extracting  the  square  roots  of 
polynomials  is  suggested  by  such  equations  as : 

(a+b)2  =  a2+2ab+b2 
(a+6+c)2=(a+6)2+2(a+6)c+c2 

=  a2+2a6+62+2ac+26c+c2 

1.  We  note  that  a2,  the  first  term  of  the  polynomial, 
is  the  square  of  the  first  term  of  the  square  root. 
Therefore,  if  the  polynomial  is  arranged  according  to 
powers  of  a  letter,  the  first  term  of  the  root  is  found 
by  extracting  the  square  root  of  the  first  term  of  the 
polynomial. 

2.  The  term  2ab,  which  is  the  first  of  the  remaining 
terms  of  the  polynomial,  is  twice  the  product  of  the 
first  term  of  the  root  by  the  next  term.  Therefore 
the  second  term  of  the  root  is  found  by  dividing  the 
first  term  of  the  remainder  by  twice  the  first  term  of  the 
root. 

3.  By  adding  b,  the  second  term  of  the  root,  to  twice  a, 
the  first  term,  and  then  multiplying  this  sum  by  the 
second  term,  b,  we  obtain  2ab-\-b2.  Subtracting  this  from 
the  first  remainder,  2ab+b2+2ac-\-2bc-\-c2,  we  have  the 
second  remainder,  2ac-\-2bc-]-c2. 


80 


THIRD-YEAR  MATHEMATICS 


By  dividing  2ac,  the  first  term  of  this  remainder,  by  2a 
we  find  c,  which  is  the  third  term  of  the  root.  The  process 
in  (3)  is  then  continued.  If  at  any  time  there  is  no 
remainder  the  polynomial  is  a  square. 


-    EXERCISES 


Find  the  square  roots  of  the  following  polynomials: 
1.  4-197*2+12?i-427i3+49n4 
First,  the  polynomial  is  arranged  according  to  powers  of  n, 
7n2—Sn—2  =  Square  root 


(7n2)5 


49n4 


-42n3-19n2+12n+4 


A2n3+9n2 


thus :  Polynomial  =  |49n4  -  42n3  -  19n2 +  12n +4 

The  first  term  of  the  root 
is  V4Qn*  =  7n2 

Subtracting  the  square  of 
7n2  from  the  polynomial, 
we  have  the  remainder: 

The  first  term  of  the  remainder 
divided  by  2  •  7n2  gives— 3n,  which 
is  the  second  term  of  the  root. 

Adding  this  to  2  •  7n2  we  have 
14n2-3n. 

This  is  multiplied  by  — 3n: 

*   (14n2-3n)(-3n)  = 

The  product  obtained  is  then  subtracted 
from  the  preceding  remainder,  giving : 

The  first  term  of  the  last  remainder  di- 
vided by  2  •  In2  gives  —2,  which  is  the  third 
term  of  the  root. 

Adding  this  to  2  times  the  sum  of  the 
first  two  terms  of  the  root  we  have 
14n2-6n-2. 

This  is  multiplied  by  -2:     (14n2-6n-2)(-2) 

The  product  obtained  is  then  subtracted 
from  the  preceding  remainder,  leaving  no 
remainder. 

Hence  the  given  polynomial  is  a  perfect  square  and  the  square 
root  has  been  found  exactly. 


-28n2  +  12n+4 


■28n2  +  12n+4 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN       81 

2.  10x2+12z3+  l+4x+9z4 

3.  16z6+10z-8x3+l-40:r4+25x2 

4.  9x2-Sxy2-30xy+5y3+25y*+^ 
|5.  x2+9z2+4y2-4xy-12yz+Gxz 

6.  z4+25+6z3-30;r-:r2 

J7.  lBa:6+25x4-24x5+10x2-20x3-4a:+l 

8.  12a-23a4+8a5+5a2+4-22a3+16a6 

o    m2_i_  1 1  ,6m  ,  6a  ,  a2 
a2  a      m     m2 

1 10.  1+z,  to  5  terms 
Solve  the  following  equations: 

11.  y2—4my+ny-\-3m2  —  5mn  —  2n2  =  0,  for  y 

12.  2/2-3ai/-3fo/+3a  =  2/-9a&,  lory 
fl3.  £2-3aZ-2  =  Z-2a2-3a,  for* 

14.  x2— Sx  —  5x2— 2mx2  =  3+mx+4m,  for  z 


Fractional  Equations 

85.  Solve  the  following  equations : 

J2.  — rr+— i—  =  s .  f or  a;     (Harvard) 

,     4a2        V        4a2-62     ,  _    _  ... 

3-5+2-^=i(i^)'f0ra;     (Sheffield) 

••^JSSSrl  (princeton) 


82  THIRD-YEAR  MATHEMATICS 


*•  ^+i=6+i=r°  (Harvard> 

„    s~3  .  2s+3     5s-3_3s-l     _ 
8-  ^i~+^ 6 2 3 

.  x+l-t-^— -  =  — —  — 

t10- 3iii)-^T+3wi)=i  (Chicag0) 


,,    9a3,,     baVa     (Za-Va\2 
1*  + ^=1 


/3a-  vVt2 


a— 26        a;      a— 2b 


Problems  Leading  to  Quadratic  Equations 
86.  Solve  the  following  problems : 

1.  A  rectangular  box  has  a  volume  of  1,500  cubic  inches. 
Its  depth  is  5  in.  and  it  is  5  in.  longer  than  wide.  Find  its 
dimensions.     (Sheffield.) 

2.  One  side  of  a  rectangle  is  20  cm.  longer  than  the  other. 
The  diagonal  is  7  cm.  longer  than  the  longer  side.  Find  the 
area  of  the  rectangle.     (Harvard.) 

%Z.  It  is  shown  in  plane  geometry  that  the  length  of  the  side, 
s,  of  a  regular  decagon  inscribed  in  a  circle  of  radius  a  is  deter- 
mined from  the  equation 

a—s_s 
s       a 

Assuming  this  equation,  solve  for  s  in  terms  of  a. 
Find  the  value  of  s  correct  to  three  significant  figures  when 
a  =  100.     (Harvard.) 

4.  On  the  side  AB  of  a  square  ABCD  a  point  E  is  marked 
at  a  distance  of  10  in.  from  A.  The  area  of  the  trapezoid  EBCD 
is  less  by  22f  sq.  in.  than  three-fourths  of  the  area  of  the  square. 
How  long  is  a  side  of  the  square  ?     (Harvard.) 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN      83 

5.  A  man,  after  having  bought  an  article,  sells  it  for  $21. 
He  loses  as  many  per  cent  as  he  gave  in  dollars  for  the  article. 
What  did  he  pay  for  it  ?     (Yale.) 

6.  A  trunk  30  in.  long  is  just  large  enough  to  permit  an 
umbrella  36  in.  long  to  lie  diagonally  on  the  bottom.  How 
much  must  the  length  of  the  trunk  be  increased  if  it  is  to  accom- 
modate, diagonally,  a  gun  4  in.  longer  than  the  umbrella? 
(Chicago.) 

7.  A  rectangular  piece  of  tin  is  4  in.  longer  than  it  is  wide. 
An  open  box  containing  840  cu.  in.  is  made  by  cutting  a  6-inch 
square  from  each  corner  and  turning  up  the  ends  and  sides. 
What  are  the  dimensions  of  the  box  ?     (Chicago.) 

$8.  An  open  box,  to  be  made  from  a  square  piece  of  card- 
board by  cutting  out  a  4-inch  square  from  each  corner  and  turn- 
ing up  the  sides,  is  to  contain  256  cubic  inches.  How  large  a 
square  must  be  used?     (Chicago.) 

9.  The  rates  of  two  trains  differ  by  5  mi.  an  hour.  The 
faster  requires  one  hour  less  time  to  run  280  miles.  Find  the 
rate  of  each.     (Yale.) 

10.  If  a  body  falls  from  rest,  the  distance,  s,  that  it  falls 
in  t  seconds  is  given  by  the  formula  s=16£2.  A  man  drops  a 
stone  into  a  well  and  hears  the  splash  after  3  seconds.  If  the 
velocity  of  sound  in  air  is  1,086  ft.  a  second,  find  the  depth  of 
the  well. 

11.  Find  the  sides  of  a  right  triangle  in  which  the  sides  of 
the  right  angle  are  respectively  20  in.  and  10  in.  shorter  than  the 
hypotenuse. 

12.  A  rectangular  tract  of  land,  800  ft.  long  by  600  ft.  broad, 
is  divided  into  four  rectangular  blocks  by  two  streets  of  equal 
width  running  through  it  at  right  angles.  Find  the  width  of 
the  streets,  if  together  they  cover  an  area  of  77,500  square  feet. 
(M.I.T.) 

|13.  What  is  the  number  of  sides  of  a  polygon  having  170 
diagonals  ? 


84  THIRD-YEAR  MATHEMATICS 

14.  Two  men  can  do  a  piece  of  work  in  6  hr.  40  minutes. 
One  can  do  the  work  alone  in  3  hr.  less  time  than  the  other.  In 
how  many  hours  can  he  do  it  alone  ? 

15.  The  sum  of  the  two  digits  of  a  given  number  is  5.  If 
the  order  of  the  digits  is  changed,  the  product  of  the  result  by 
the  original  number  is  736.     Find  the  number. 


Equations  of  Quadratic  Form 
87.  Solve  the  following  equations : 
1.  2/4-262/2+25  =  0  |4.  6*4+6=13*2 


2.  z6-8  =  7z3  6.  (z2-2z)2-7(z2-2z)  +  12  =  0 

t=97 
2     9 


4     Q7 
3.  W+-,  =  -k  $6.  (4z+5)2+2(4:c+5)-15  =  0 


Trigonometric  Equations 

88.  Conditional  equations  containing  trigonometric 
functions  of  unknown  angles  are  called  trigonometric  equa- 
tions.    For  example,  2  sin  x  =  tan  x,  5  sin2  x-\- cos2  x  =  2. 

Some  trigonometric  equations  are  easily  solved  by 
factoring. 

If  the  equation  contains  several  trigonometric  func- 
tions of  x}  it  is  generally  best  to  change  its  form  so  that 
it  contains  only  one  function  of  x.  This  is  accomplished 
by  use  of  the  following  fundamental  identities : 

.,  1  _  cos  X 

1.  sin  #  csc  £=1  5.  cot  x=- 

sin  x 

2.  coszsec#s=l  6.  sin2  £+cos2  z=l 

3.  tan  x  cot  x=l  7.  tan2  #+l  =  sec2  x 

4.  tan  z= 8.  cot2  z+l  =  csc2  x 

cos  x 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN      85 


EXERCISES 

Find  all  the  positive  values  of  the  angle  between  0°  and  360c 
which  satisfy  the  following  equations : 

1.  cot  x  =  2  cos  x 

CO* 


a.                   cos  x 
Since  cot  x  = 


sin  x 
cos  x 


sin  x 

Hence,  —2  cos  x=0 

sin  x 

Factoring,        cos  x  (  — 2 )  =  0 

Vsin  x      / 

This  equation  is  satisfied  if  cosx  =  0 


=  2  cos  x 


Fig.  56 


or  if 


2  =  0 


sin  x 

cos  x  =  0,  x  is  equal  to  90°,  or  270c 

— 2  =  0,  then  sin  x  =  § 


.*.     x  =  30°,  or  150°,  Fig.  56. 
.*.  x=30°,  90°,  150°,  and  270°  are  the  positive  values  of  x,  less 
than  360°,  satisfying  the  given  equation. 

2.  5  sin2  x+cos2  x=    f 

Since  sin2  x  =  1 —cos2  x,  it  follows  that 

5  —  5  cos2  x+cos2  x  =  2 
.'.    4  cos2  x  =  3 

cosx  =  =|"/3. 
If  cos  x  =  +  |t/3,  it  follows  that  x  =  30°,  or  330°. 

If  cos  x  =  -  Jt/3,  it  follows  that  x  =  150°,  or  210°. 

.'.  x  =  30°,  150°,  210°,  and  330°  are  the  positive  values  of  x,  less 
than  360°,  satisfying  the  given  equation. 

3.  tan  x  sec  x—  V2 

1 


Put 


sin  x 

tan  x  = ,  sec  x 

cos  x 


cos  x 


,  cos2x  =  l— sin2x 


Then  show  that  = r-r—  =  V2 

1—  sin2  x 

.'.    sin  x  =  V2  —  V2  sin2  x 

Solve  this  equation  for  and  find  the  required  values  of  x. 


86  THIRD-YEAR  MATHEMATICS 

4.  2  cos2  x+3  sin  z-3  =  0  9.  2  sin2  0+3  cos  0=0 

5.  3  sec2  x  —  7  tan2  x  =  tan  a;         +</>         „  .      .     „     1 

$10.  cos2  0-sin  0  =  t 

6.  4  cos2  z  =  cot  x 

7.  tanz  =  cosz  11.  tan  0+cot  0  =  2 
$8.  sin2  0-cos  0+1  =  0                  $12.  sin2  0-2  cos  0=^ 

Nature  of  the  Roots  of  a  Quadratic  Equation 

89.  Complex  numbers.  We  have  seen,  §82,  that 
some  quadratic  equations  cannot  be  solved  by  graph,  e.g., 
the  equation  x2— 4a;+5  =  0.  However,  by  means  of  the 
quadratic  formulas  we  find  that 

4±i/l6-20    4±l/^4     rt       , 

*  —    -^ —J—  =2-l/-l 

Thus  the  roots  of  the  equation  £2  —  4z+5  =  0  involve 
the  square  root  of  a  negative  number.  We  cannot  extract 
the  square  root,  or  any  even  root,  of  a  negative  number, 
since  the  square  of  a  real  number  is  always  positive.  Thus, 
l/-4  cannot  equal  +2  or  -2,  since  (+2)2=  (-2)2= +4. 

An  even  root  of  a  negative  number  is  called  an 
imaginary  number. 

Expressions  of  the  form  of  a+V^b,  where  a  is  a  real 
number  and  b  a  positive  real  number,  are  complex  num- 
bers. Thus,  l  +  l/^2,  i/+3_i/38  are  complex  num- 
bers. They  are  also  called  imaginary  numbers.  However, 
the  latter  term  is  misleading,  because  complex  numbers 
are  not  imaginary  for  the  person  who  has  made  a  study 
of  these  numbers. 

90.  Classification  of  numbers.  Positive  integers  and 
fractions  are  the  first  numbers  with  which  the  pupil 
becomes  acquainted.  Later  the  study  of  negative  num- 
bers is  taken  up.     Positive  and  negative  integers,  the 


GASPARD  MONGE 


GASPARD         MONGE 

GASPARD  MONGE,  the  son  of  a  small  peddler,  was  born 
at  Beaune  in  1746  and  died  at  Paris  in  1818.  A  plan  of 
his  native  town,  drawn  by  him,  fell  into  an  army  offi- 
cer's hands,  and  its  excellence  so  impressed  the  officer  that  he 
recommended  that  Monge  be  admitted  to  the  training  school 
at  Mezieres.  His  low  birth  prevented  him  from  receiving  a 
commission  in  the  army,  but  he  was  allowed  to  attend  the 
annexe  of  the  school,  where  he  learned  surveying  and  drawing. 
But  he  was  not  sufficiently  well  born  to  be  allowed  to  do  calcu- 
lator problems.  A  difficult  plan  for  a  fortress  was  to  be 
drawn,  and  Monge  did  it  by  a  geometrical  construction.  This 
turned  the  tide  of  young  Monge's  fortunes.  The  officer  at 
first  objected  to  receiving  Monge's  plan  because  he  had  taken 
less  time  than  etiquette  required  for  such  a  problem,  but  the 
superiority  of  Monge's  method  finally  won  its  acceptance. 

In  1768  Monge  was  made  professor  of  descriptive  geom- 
etry, though  the  results  of  his  methods  were  to  be  a  secret 
confined  to  officers  above  a  certain  rank. 

In  1780  he  was  made  professor  of  mathematics  at  Paris, 
and  he  communicated  his  earliest  paper  of  importance  to  the 
French  Academy  in  1781.  The  paper  discussed  lines  of  curva- 
ture drawn  on  a  surface.  He  found  that  the  validity  of  solu- 
tions is  not  impaired  when  imaginaries  are  involved  among 
subsidiary  quantities.  Euler  had  treated  these  questions,  but 
Monge's  methods  were  superior  to  those  of  Euler.  Monge 
applied  his  results  to  central  quadrics  in  1795.  In  1786  he 
had  published  a  work  on  statics. 

Monge  became  embroiled  in  the  politics  of  the  Revolution 
and  narrowly  escaped  the  guillotine.  In  1798  he  was  sent  to 
Italy  on  state  business,  and  thereafter  joined  Napoleon  in 
Egypt.  After  Napoleon's  defeat,  he  escaped  to  France  and 
settled  down  at  Paris. 

Monge  was  now  made  professor  and  gave  lectures  at  the 
Polytechnic  School  of  Paris  on  descriptive  geometry,  and  in 
1800  published  his  text  entitled  G6om4trie  descriptive.  In  this 
he  treats  the  theory  of  perspective  and  the  theory  of  surfaces 
in  a  masterly  way. 

On  the  restoration  he  was  deprived  of  his  offices,  stripped 
of  his  honors,  and  thrown  out  of  the  French  Academy.  These 
humiliations  soon  led  to  his  death. 

[See  Ball,  pp.  426-27,  and  Cajori,  pp.  286-87.] 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN      87 

fractions  and  zero,  form  the  domain  of  rational  numbers. 
For  example,  4,  —8,  -§,  —1.75,  are  rational  numbers. 
A  rational  number  may  always  be  expressed  exactly  as  the 
quotient  of  two  integral  numbers.  This  includes  integers, 
since  they  are  fractions  with  1  as  denominator. 

However,  such  numbers  as  Vz,  V^  3^7,  cannot  be 
expressed  exactly  as  quotients  of  integers.  They  are 
classed  as  irrational  numbers.  The  rational  and  irra- 
tional numbers  form  the  domain  of  real  numbers.  For 
example,  the  number  t,  known  to  us  from  the  study  of 
the  circle,  is  a  real  number,  but  not  a  rational  number,  as 
it  cannot  be  expressed  exactly  as  a  quotient  of  two  integral 
numbers. 

91.  Graphical  representation  of  real  numbers.     Posi- 
tive integers  may  be  represented  by  equidistant  points 
on  a  straight  line,  as  OA,  Fig.  57, 
or  by  the  distances  of  these  points       c  -,'-;»-,' -/  °  ),*  ;  ;   , 
from  a  fixed  point,  as  0.  B  o  c  A 

Negative  numbers  are  then  Fig.  57 

represented  by  equidistant  points 
laid  off  in  the  direction  opposite  to  that  of  OA,  as  OB. 

The  origin,  0,  represents  zero. 

Fractions  are  represented  by  intermediate  points. 
Thus  the  point  C  represents  the  fraction  -|.  Similarly 
any  rational  number  may  be  represented  by  a  point  on  the 
line. 

Although  between  any  two  rational  numbers,  however 
close,  other  rational  numbers  may  be  inserted,  there 
are  points  on  the  line  which  do  not  represent  rational 
numbers.  For  example,  we  know  that  V~2  is  the  length 
of  the  diagonal  of  a  square  whose  side  is  of  unit  length. 
Therefore,  by  laying  off  on  OA  a  length  equal  to  this 
diagonal,  we  obtain  a  single  definite  point  which  represents 


88  THIRD-YEAR  MATHEMATICS 

the  irrational  number  l/2.  Indeed  it  can  be  shown  that 
to  any  irrational  number  there  corresponds  a  definite 
point  on  OA. 

Hence  any  real  number  can  be  represented  by  a  point 
on  a  straight  line.  This  line  is  called  the  axis  of  real 
numbers. 

92.  Nature  of  the  roots  of  a  quadratic.  The  char- 
acter of  the  roots  of  the  equation  ax2-{-bx-\-c  —  0  depends 
upon  the  number  b2—\ac.  The  function  b2— 4ac  is 
called  the  discriminant  of  the  equation  ax2+bx-\-c  =  0. 
In  the  following  we  consider  the  coefficients  a,  b,  and  c 
to  be  rational  numbers. 

1.  If  b2— 4ac  =  0,  the  two  values  of  x  are  the  same. 
Thus  the  roots  of  the  quadratic  are  real,  rational,  and 
equal. 

For  example,  for  the  equation  x2— 6z+9  =  0  we  have 
62-4ac  =  36-36  =  0. 

Hence,  without  solving  the  equation,  we  know  that 
the  roots  are  real,  rational,  and  equal. 

2.  If  b2— 4ac>0,  the  roots  are  real  and  unequal. 

(1)  If  62— 4ac  is  a  square,  the  roots  are  rational. 

(2)  If  62— 4ac  is  not  a  square,  they  are  irrational. 
For  example,  for  the  equation  x2— 92+14  =  0  we  have 

6*-4ac  =  81  -56  =  25. 

Hence  the  roots  are  real,  rational,  and  unequal. 
For  the  equation  x2+5x+l  =  0  we  have 
62-4ac  =  25-4  =  21. 

Hence  the  roots  are  irrational. 

j  

3.  If  b2— 4ac<0,  the  expression  Vb2— 4ac  is  imaginary 
and  the  roots  of  the  equation  are  called  complex. 

Thus  for  2x2+z+l  =  0  we  have  fc-4ac=l-8=  -7. 
Hence  the  roots  of  this  equation  are  complex. 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN       89 


The  preceding  discussion  may  be  represented  in  a 
table  as  follows : 


The  roots  of  a 
quadratic  equa- 
tion are 


complex,  if 
62-4ac<0 


real,  if  b2  — 4ac 
is  not  <0 


rational 
and 


equal,  if 
&2-4ac  =  0 

unequal,  if 
62-4ac>0 
and  a  perfect 
square 


irrational,     if 
62-4ac>0, 
and  not  a,  per- 
k  feet  square 


EXERCISES 

Without  solving  determine  the  nature  of  the  roots  of  each 
of  the  following  equations : 

1.  3z2-8z+5  =  0* 

62-4ac  =  64-4(3)(5)  =64-60  =  4 
.*.  the  roots  are  real,  rational,  and  unequal. 

2.  x2-4x+8=--0  6.  9x2+12x+4  =  0 

3.  a2+3a-l  =  0  7.  7y2+3y  =  0 

4.  5z2-3z  =  2  8.  5z2+7:c+3  =  0 

5.  3z2  =  7z+6  9.  z2-6z+4  =  0 

Find  the  values  of  d  for  which  the  roots  of  the  following 
equations  are  equal: 

10.  9x2+(l+d)x+4:  =  0 
62-4ac  =  l+2d+d2-144 
Hence,  to  make  the,  roots  equal,  we  put 
d*+2d- 143=0 

.\d-ll,  -13 


90  THIRD-YEAH  MATHEMATICS 

11.  y*+y+d=0  U4.  y2+3dy+d+7  =  0 

12.  2x2+(l+d)x+2  =  0         15.  (d+l)x'i+dx+d+l  =  0 

13.  z2-4dz+4=0  |16.  2dx2+(5d+2):r+(4d+l)  =  0 

Relation  between  the  Roots  and  the  Coefficients  of  a 
Quadratic 

93.  Denoting  the  roots  of  the  equation  ax2+bx+c  =  0 

by  

-  -  -b+VV-±ac  and  r2=  -b-VW-iac 


2a  2a 

we  have  by  addition : 

-2b        b 

by  multiplication : 

(-6)2-(j/fr2-4ac)2_62-62+4ac^c 
2a  •  2a  4a2  a 

b 


n  •  r2 


Hence  the  sum  of  the  roots  is  — 

a 


and  the  product  of  the  roots  is 


EXERCISES 

Find  the  sum  and  the  product  of  the  roots  of  the  equations 
in  exercises  1  to  5. 
1.  2x2-9z+8  =  0 
Since  a  =  2,  b=  —9,  c  =  8,  it  follows  that 

-9    9 

ri+r2=__  =  - 

and  that 

8       A 

nr2  =  2  =  4. 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN      91 

2.  2x2-9x-5  =  0  4.  x2+2x+2  =  0 

3.  x2-12x-13  =  0  5.  4-?/-6?/2  =  0 

6.  One  root  of  the  equation  x2— kx+2l  =  0  is  7.     Find  the 
value  of  k. 

n  •  r2  =  21 
n       =7 
.'.  r2  =  3  and  A;  =  10 

7.  One  root  of  the  equation  Sx2— kx -f  10=0  is  5.     Find  k 
and  the  other  root. 

8.  Find  the  values  of  p  and  q  in  the  equation  x2-\-px+q  =  0, 

1.  If  the  roots  are  6  and  —4, 

2.  If  the  roots  are  S-V&  and  3+  /6. 

9.  Form  the  equation  whose  roots  are  2  and  —3. 

b  c 

Since  2  + (—3)  =  —  and  2(— 3)=~,  we  may  substitute  these 

Cb  Qi 

b       c 
values  in  the  equation  x2-\ — x-\ —  =0. 
a      a 

This  gives  z2+z-6=0. 

10.  Form  the  quadratic  equations  whose  roots  are : 

1.  -3,10  5.     2=^3 

2.  -8,  -3  6.     a,  -b 

3.  6,  -i  7.     Vd,  -VI 


0 

2>  4 


4.    i,  x  8.     — ra-f-n.  —m—n 


Factoring 

94.  The  solution  of  a  quadratic  equation  enables  us 
to  find  the  factors  of  the  quadratic  trinomial  ax2+bx-\-c. 


Show  that  ax2+bx+c  =  a[  x2-\ — 


a      a/ 


=  a[x2-  (ri+r2)z+rir2] 
,\  ax2+bx+c=a(x— ri)(x-r2). 


92  THIRD-YEAR  MATHEMATICS 

EXERCISES 

Determine  the  factors  of  the  quadratic  functions  given  in 
exercises  1  to  7. 

1.  5x2-h3z-20 
Let5x2+3x-20  =  0 

Tum  ,      -3  +  ^409      r      -3-/409 

-3-/4091 


5x*+3x-20^[x-=^™\ 


10        J 

2.  2x2+5x-7  5.  z2-2ax+a2-62 

3.  2x2+5x+2  |6.  z2+4a6x-(a2-62)2 

4.  Gy2+y-l  J7.  abx2±(a+b)x+l 

Reduce  the  following  fractions : 

2?/2+7?/-4  .         ax2-2ax+x+a-\ 

'  3y2+lh/-4  *     '  ax2-3ax+x+2a-2 

2t2+8t-90  .        20xhf+20xyz-21z2 

'  3£2-36£+105  l    '  10x2?/2+24:n/z-18z2 

Summary 

95.  The  chapter  has  taught  the  meaning  of  the  follow- 
ing terms : 

quadratic  equation  real  numbers 

normal  form  of  a  quadratic  rational  numbers 

equation  irrational  numbers 

imaginary  numbers  nature  of  the  roots  of  a 
complex  numbers  quadratic,  discriminant 


QUADRATIC  EQUATIONS  IN  ONE  UNKNOWN      93 

96.  The  following  are  typical  problems  of  the  chapter : 

1.  To  solve  quadratic  equations  in  one  unknown  by 
the  following  methods : 

(1)  By  factoring, 

(2)  By  formula, 

(3)  By  completing  the  square, 

(4)  By  graph. 

2.  To  extract  the  square  root  of  a  polynomial. 

3.  To   solve   fractional   equations   which   reduce   to 
quadratic  form. 

4.  To    solve    equations    of    other   degree   than   the 
second,  but  of  the  form  of  quadratics. 

5.  To  solve  trigonometric  equations. 

6.  To  solve  problems  leading  to  quadratic  equations. 

7.  To  represent  real  numbers  graphically. 

8.  To  determine  the  nature  of  the  roots  of  a  quadratic 
equation. 

9.  To  solve  quadratic  equations,  having  given  a  rela- 
tion between  the  roots  and  the  coefficients. 

10.  To  factor  quadratic  trinomials  by  means  of  the 
quadratic  formula. 


CHAPTER  V 
FACTORING.    FRACTIONS 

97.  Purpose  of  the  chapter.  The  work  in  factoring 
given  in  §§22  and  94  completes  what  is  commonly  taught 
about  this  subject  in  an  elementary  course  in  mathematics. 
We  have  had  no  general  method  by  which  all  polynomials 
can  be  factored,  but  we  have  learned  to  recognize  certain 
forms  which  suggest  a  particular  method  to  be  used  in 
factoring  them.  It  is  the  purpose  of  this  chapter  to  review 
and  summarize  what  we  should  know  about  factoring 
and  to  make  use  of  factoring  in  the  operations  with  alge- 
braic fractions. 

98.  The  difference  of  two  squares. 

Find  the  prime  factors  of  the  following: 

i.  x2-±y2  7.  16a2-(2m-3w)2 

x2-4y*  =  (x+2y)(x-2y) 


2.  9a2 -25 

3.  1-r4 


8.  {x2-y2)2-x* 

9.  9c2- (3a -2c)2 

10.  36(a+6)2-25(c-rf)2 


4.  9a2x«-Wy2  %lm  (a+6)2_(a_6)I 

5.  a6-66,  a8-68  12.  (4a+5)2-(2x-3)5 


)2 
6.  a4-64,  a12-b12  13.  (x+y+z)2-(x-y-z)2 


Reduce  the  following  fractions  to  simplest  terms : 

m{x2-y2)  x2+y> 

14*  (a+b)(x-y)  16*  x*-y* 

(x+a)(x-b)  (4a-d)2-(26-3c)2 

°'  (x2-a2)(x2-b2)  17*  (4a-26)2-(3c-d)2 

94 


FACTORING.    FRACTIONS 


95 


Add  and  subtract  as  indicated : 
x  x      .  2x-6 


18 


19.  3x+ 


3x2 


2x-3    2x+3  '  9-4x2  **"  w  '  x-3    x+3  '  x2-9 

Multiply  and  divide  as  indicated : 


20 


(ft+8)2    a2- 16 
(ft-4)2 '  ft2-64 


21. 


x2-9     (x+3)2 


x-4  *  (x-4)2 

99.  The  sum  or  differences  of  like  powers. 

Factor  the  following : 

1.  64ft3+2763 

64a3+2763  =  (4a +36)  (16a2  -  12a6+962) 

2.  8x3+125v3  7.  8vis+27wis 

3.  27x3-8?/3  8.  (a+&)3-c3 

4.  343+x3  9.  (w+3)3+ft3 

5.  pz— J  10.  x3— (m+n)3 

6.  512c3+27d3  11.  (5m-7i)3+c3 
Reduce  to  the  simplest  form : 


12 


1 


14 


x2+x+l 
Multiply  as  indicated : 

X2  __yi  x6_  y6 


X? 


16 


x3— if     xz-\-yz     x2+?/2 

Subtract  as  indicated  : 
a2+ft        1 


a3-l     a-1 

Solve  for  x: 
2x-3  .     x 


13. 


15. 


17. 


64a666+l 
4a262+l 

x2— 9     x 

2+3x+9 

x3-27 

x+3 

3       5 
a    a— 1 

2ft-3 
ft2-l 

18. 


x2-l  '  x-1    x+1 
Factor  the  following: 
19.  ft6+66;  ft9-69;  a9+69;  ft12+612 


96  THIRD-YEAR  MATHEMATICS 


100.  Trinomials. 

Factor  the  following : 

1.  4x2-12xy+9y2 

7.  x*-3x2y2+yA 

2.  x*y2+2x2yz+z2 

8.  a4+2a262+964 

3.  5x2-38:r+21 

9.  6z2-17x+5 

4.  662-296+35 

10.  3z2+8z-7 

5.  1  —  6xy+5x2y2 

11.     (<U  +  V)2  +  4:t(U  +  V)+4:t2 

6.  2z2+llz+12 

12.  9&2+6&(r+s)  +  (r+s)2 

Reduce  the  following 

fractions  to  lowest  terms : 

z2-3x+2 
13.     o     ..     ,  o 

2a2+17a+21 

16.    o    9  i   oc      i   or 

z2-4x+3  3a2+26a+35 

232+5s+3  2m2+5m+2 

3x2+7x+4  2m2+7m+3 

Perform  the  indicated  operations : 
62+&-6       &2+8&+15 


17. 
18. 
19. 


262+56-3  262+76-15 
106£+3&2+3s2^  (3b+x)x 
10bx-3b2-3x2''  (x-3b)b 
4a3-9a    a2-2a-35 


a-7         2a3-3a2 

Solve  the  following  equations : 

_2 3-4s_  3 

20*  3+2x    9-4x2    9-12x+4x2 

2        2-s  _       2+s 
21*  z-f4+z2-16    z2-8z-r-16_0 

101.  Polynomials. 

Factor  the  following  polynomials : 

1.  16a262+48a2&-16a6+8a  3.  x*+x*-x-l 

2.  6ax2y-Sax*-3ay2-4;axy  4.  a2+62-c2-2a& 


FACTORING.    FRACTIONS 


97 


5.  a2-b2+a+b 

6.  xA+4x3-8x-32 

7.  ra2+6m-z2+9-4z?/-4?y2 

8.  12y3+3y2-8y-2 

9.  a3+63+a+6 

10.  x2y-\-y2z-\-xz2— x2z— xy2— 

11.  8x*+36x2y+54xy2+27y3 

Reduce  to  lowest  terms : 

x2-\-ax-\-bx-\-ab 


15 


16. 


(z2-a2)(a;2-62) 

x2+ax-\-bx+ab 

x2+3ax+2a2 


12.  3x3-2x2+5x-6 

yz2 ..  13.  x3+9z2+10:r+2 

14.  z3-6:r2+lia;-6 

2az  —  6x — 2^7/ + 6?y 
3a/cz  —  9  kx  —  3aky + 9ky 
2ax+3&?+4a+6& 


17. 


18. 


z2+z(6+2)+26 


Perform  the  indicated  operations : 


19. 


20. 


21. 


x2-\-y2-\-2xy—z2_z_x+y-\-z 
z2—x2—y2-\-2xy  '  y+z—x 
3a— 36      cx—dx-\-cy—dy 


5c  —  5d    am—bm+an—bn 

l + I 

ab-\-ac  —  bk  —  ck    bm+cm-\-bk+ck 


Miscellaneous  Exercises  on  Factoring 
102.  Factor  the  following : 


1.  x5-l 

10.  (x2-5x)2-2(z2-5z)-24 

2.  16x2+252/2+40a;i/ 

11.  z(x+l)(4x-5) -6(3+1) 

3.  6x2+llx-10 

12.  4x4+y4-5x27/2 

4.  z4+x2?/2+?/4 

13.  a4-14a262+64 

6.  x4-\-xzy—xy3—y4 

14.  64x6a-7/12a 

6.  x3-7x+6 

15.  a2+62+2a&+8a+86-9 

7.  a3c3+63 

16.  a4+4 

8.  32a2-29a&+562 

17.  a4-a262+62-l 

9.  ~-3^+2 
y2      x2 

18.  (a+6)2-(a2-62)-6(a-6)2 

98  THIRD-YEAR  MATHEMATICS 

Miscellaneous  Exercises  on  Fractions 
1  J103.  Solve  the  following: 

1.  Simplify: 

2.  Simplify: 

\    a3c3       Icai  \   '    I  c2     a2  \ 

Check  by  substituting  a  =  2,  c=l,  in  the  original  fractions 
and  in  the  result.     (Sheffield) 

3.  Simplify: 


rrf+mn    mz— mn2  — m2-\-n2    m2n2+??m3+n4 
m2+w2  m3n— n4  m4— 2m3+m2 

.  ra3n+2m2n2+ran3 


(Chicago) 


(Chicago) 


m4— ?i4 

4.  Subtract  as  indicated: 

r-\-s  s-M  r-\-t 

(r-t)(s-t)     (r-s)(t-r)     (t-s)(s-r) 

5.  Reduce  to  the  simplest  form: 
a2-4ax-2lx2  ax+2by+2bx+ay 

a2-49x2  x3+3x2y+3xy2+y* 

bam  +1  Pan  1  1 

2ra2+5raw+2n2  xy-\-y2    x2+xy 

(a2-c2)x2+2ax+l  a*-2a2-a+2 

ax— cx-\-l  a2— 1 

(X-y)(X-Z)  +  (y-X)(y-Z)  +  (Z-X)(Z-y)       (ChlCaS°) 

6.  Simplify: 


f     rg-l  +  l/g-l_(g-2)(g-3K-i 
x     L    2        2U+1         x(x+l)      /J 


(Princeton) 


7.  Simplify: 

(t^+—  )+(t£—  —  )     <Harvard> 

\l-\-x       x   /      \l-\-x       X    / 


FACTORING.    FRACTIONS  99 

8.  Simplify: 

(«2+^)(^Ha-f6+i)     (ShefMd) 

9.  Reduce: 

ffiiwSS  (SheffieId) 

10.  Reduce: 

1+*-*-*+*=*    (Sheffield) 
1— a2  a-5  — 1 

11.  Simplify: 

(■-s=5H«:5=2)  <** 

12.  Simplify: 

13.  Simplify: 

8c3-l         A        4    \  .  /2c-l\     /15       ,, 
9c2-12c+4-(1-^+2)-(9c^i)     (B°ard) 

14.  Simplify: 

ja^_^+ _M.U  ^-l)     (Board) 

15.  Find  the  highest  common  factor  and  the  least  common 
multiple  of  x3—3x2+x— 3      ,_  .      ... 

and  s*-3s*-s+3      (Columbia) 

Miscellaneous  Exercises  on  Complex  Fractions 
£104.  Solve  the  following : 
1.  Simplify  the  following  expression: 
a    _x—a 

(Harvard) 


1    2,     \ 


x+a    x2—a? 


100  THIRD-YEAR  MATHEMATICS 

2.  Reduce  the  following  expression  to  a  single  fraction; 


U ^7T    (Harvard) 


3.  Solve: 


/1+x     l-x\  /3.x      \     r      "h2x-6/2X     ,v  ,  N 
(l^-I^)(^+-4-^)=  J5_         (YalG) 

x— 3 

4.  Reduce  to  the  simplest  form: 


11     .     2a 


a-\-x    a—x    a2— x2     /rn.        N 
1  1  2a       (ChlCag0) 

a-\-x    a—x    a2—x2 

5.  Reduce: 

(Chicago) 


a2— 2ax  ,  1 
x2 

6.  Reduce: 


2+acH — 


1— x     \-\-x  t    ac 

~~2~       ;  a2c2-l 
1-z4 

7.  Simplify: 


(Chicago) 


W-pl .  £J  A     J_\     (Princeton) 
1-2/2  x2-z  i_l 


x, 


FACTORING.   .FRACTIONS  401 

8.  Simplify: 

6a3+7a62+1263  1  fTX  ,, 

3a2-5a6-462  "  3      5a+46     (HarVard) 
196       19a2 

9.  Solve: 

_5         7_ 
X    3    ,     3    *  1        +1 


5 
3 

10.  Solve: 


(s-1)    |(1+*)    35(l-J)     7 


1+-^-= 2 .    (Harvard) 

z+2aH — 
a; 

11.  Solve  for  z: 

,       2 

12.  Show  that  the  equation 

2ax  1 


x+2a2+6a  s         ,  a+3 

a2+£2-9       x 
reduces  to 

x2(z2-2ax+a2-9)   =0    (Harvard) 

13.  Simplify: 

% s      |     %2  /i.^Wl-^l 

x+2y    2y-x^x*-4y\  \       9/     \       16/     (YaJe) 
4y-:c                 '  t     7c4 

(2?/-a;)2  144 

14.  Simplify: 

g2+&2_ 

6 e  a2— 62  ^  /a+6,  a— 6\  e  /   a     ,     b   \ 

1_1    "  a^+F3 '  U^    a+6/  *  \a+&    a-b) 
b    a 


102. 


THTTlDiYE^  .MATHEMATICS 


15.  Simplify: 


a+- 


16.  Simplify: 


x-l\2 


/x+l\2_/x  —  l\' 
\x-l)      Wu 


(Cornell) 


I.  Binomials 


Summary 

105.  Polynomials  to  be  factored  may  be  classified 
according  to  the  number  of  terms  they  contain. 

1.  The  difference  of  two  squares: 

(1)  x2-y2=(x+y)(x-y) 

(2)  (a+b)2-(s+t)2 
=  (a+b+s+t)(a+b-s-t) 

2.  The  difference  of  two  cubes: 
x*  -  y*  =  (x  -  y)  (x2+xy+ y2) 

1 3.  The  sum  of  two  cubes: 

x* + yz  =  (x + y)  (x2  -  xy + y2) 

4.  The  sum  or  difference  of  like  powers  higher 
than  the  fourth: 

x5—yb;  x5-{-y5;  x7—y7;  x7-\-y7;  etc. 

1.  The  trinomial  of   the  form   ax2-\-bx-\-c 
Factored  by  trial: 

(1)  10x2-17:c+3  =  (2z-3)(5:c-l) 
Factored  by  formula: 

(2)  ax2-\-bx-\-c  =  a(x— n)(x— r2),nandr2 
being  the  roots  of  the  equation 
ax2+bx+c  =  0 

|2.  The  trinomial  square: 
x2±2xy+y2=(x±y)2 

3.  The  incomplete  trinomial  square: 
z4+a;y+y4=  (xA+2x2y2+y4)-x*y2 

=  (x2+y2+xy)  {x2+y2-xy) 


II.  Trinomials 


FACTORING.    FRACTIONS 


103 


III.  Polynomials, 
not  including 
the  forms 
given  in  I  and 
II 


11.  Polynomials      containing      a      common 

monomial  factor: 

ax-\-ay-{-az  =  a(x-\-y-\-z) 
2.  The  perfect  cube  of  a  binomial: 

a3±3a2&+3a&2=*63=(a±6)3 
13.  Polynomials    whose    terms    may    be    so 

grouped  as  to  change  them  to  one  of  the 

preceding  forms: 

(1)  x2+2xy+y2-k2=(x+y)2-k2 

(2)  x2+2xy+y2-a2+2ab-b2 
•      =  {x+y)2-{a-b)2 

(3)  x2+2xy+y2-5x-5y+6 
=  (x+y)2-5(x+y)+6 

4.  Polynomials  containing  binomial  factors 
of  the  form  x±a: 
3x3-x2-4x+2=(x-l)(3x2+2x-2) 


CHAPTER  VI 

EXPONENTS.    RADICALS.     IRRATIONAL  EQUATIONS 

The  Fundamental  Laws  of  Positive  Integral  Exponents 

106.  Base.  Exponent.  Power.  The  symbol  a3  means 
a  •  a  •  a.     Similarly,  an  means  a  •  a  •  a  •  a. . .  .(n  factors). 

The  number  a  in  an  is  the  base,  n  is  the  exponent, 
and  an  is  the  nth  power  of  a,  or,  briefly,  an  is  a  power. 
Accordingly,  an  has  a  meaning  only  if  n  is  a  positive 
integer. 

107.  The  product  of  two  powers  having  equal  bases. 

The  product  of  two  powers  having  the  same  base  may  be 
simplified  as  shown  in  the  following  illustrations : 

1.  53-  54=(5-  5-  5)(5-  5-  5-  5)  =  57 

2.  (-2)2.  (_2)3  =  (-2)(-2)(-2)(-2)(-2)  =  (-2)* 

3.  am-an  =  [a  •  a-  a. . . .  (m  factors)]  [a  •  a  •  a (n  factors)] 

=  [a  •  a  •  a  •  a. . . .  (m-fn)  factors] 

=  am+n 

.     Qm  ,  an_am+n 

Express  this  law  in  words. 

EXERCISES 

Write  the  following  expressions  in  simplest  form: 

1.  a9  •  a  8.  (2a)3  •  3a2 

2.  (-a)9-  (-a)  9.  (-6)3-(-6)5 

3.  fc2<*  •  6«  10.  (-2)3+(-5)2-(-l)4 

4.  a**1  •  a2  11.  2(x+?/)2  •  S(x+y)3 

5.  z3r  •  xr  12.  («-&)"  •  (a-6)2"+3 

6.  axbx  •  a2*63*  13.  3a2w+36n~4  •  4an"663n-2  •  a?bAn 

7.  2a3  •  3a2  14.  a\a+b)\a-b)r-1  •  a^a+fc)'*1 

104 


EXPONENTS.     RADICALS.     IRRATIONALS        105 

Find  the  value  for  x  =  —2: 

15.  3z3-2z2+5£-4  16.  rf+2x?-7x*-Sx+2 

Factor  the  following  polynomials: 

17.  x4m-5x2m+6  18.  xm+i+2xm+2+10xm 

108.  The  quotient  of  two  powers  having  equal  bases. 

The  following  examples  illustrate  how  to  find  the  quotient 
of  two  powers  with  equal  bases : 

a5 _ft  •  ft  •  ft  •  a  •  a _  2 
'a3    ft  •  &  •  & 

o    (~3)6    (-3)(-3)(-3)(-3)(-3)(-3) 

z"  (-3)4~(-3)(-3)(-3)(-3)  ~{    6) 

am    ft  -ft  -ft  -^....(m  factors) 

3.       —^-  =  - — : — : — : ? — * — i — s==a*a' — [{m—n)  factors] 

an     ft  •  ft  •  ft  •  ft (n  factors)  LV  J 

=am~n 


a1 


=  am   n;  provided  m>n. 


Express  this  law  in  words. 

EXERCISES 

Write  the  following  expressions  in  simplest  form: 

1.  x10+x  a2  •  a2n~l 

7. 


m 


12 


an 


2*   m2  (a2-62)(s3+i/3)2 

3.  (_a)6^_(_a)2  #      (x+y)2(a+b)2 

ax+4  x2y2{x-y)A 

4'  -tf-  y'  (x»-^)(x-y)» 

(a+5)3x+y  9a663c4^  3a26c3 

5-   (a+5)x+j/  10*  4zyz  :  Sxby2z3 

a2n  a^V^+^V3 

6.     it  11. 

a2  z?/;2 


106  THIRD-YEAR  MATHEMATICS 

109.  The  power  of  a  product.     The  following  illustra- 
tion shows  how  to  find  the  power  of  a  product : 

(ab)z  =  ab  •  ab  -  ab 

=  a  • a • a • b • b  •  b 
=  a363 

Similarly,  show  that  (2  •  5)3  =  2353;   (3  •  a)4  =  34a4. 

Show  that  (ab)m=ambm,  or  ambm=(ab)m. 

Express  this  law  in  words. 

EXERCISES 

Express  the  following  powers  as  products : 


1.  (2ab)A                         3.  (Sxyzy 

5.  (abxy)2n 

2.  (xy)z                           4.  (-2ab)2 

6.  (2ran  •  Sp): 

Find  the  value  of  each  of  the  following: 

7.  23  •  33                         8.  22  •  52 

10.  202-52 

23.33  =  (2.3)3                 43.253 
=  63=216            9-4'^ 

11.  34-24 

110.  The  power  of  a  quotient.    The  following  illus- 
tration shows  how  to  find  the  power  of  a  quotient : 

/aV  — a    a    a_fl3 
W  ~b'b'b=b3 

Similarly,  show  that: 

V  ~V;   W  ~(3?/)4 
Show  that         (°)    -£*;    or   £*«(£) 
Express  this  law  in  words. 


EXPONENTS.     RADICALS.     IRRATIONALS        107 

EXERCISES 

Express  the  following  powers  as  quotients: 

1    {W-Y  q    /lOaby  _    f-2mty 

X'  \2abJ  3*  \9acJ  5*  \-5pq) 

/gy\4_  (ay)4      sV 

V2a6/       (2a&)4     16a464 


*  (*)'      «■  (- 

.CO  I"* 

r    /(2x)(3y)' 
6'  V(4a)(-6). 

Find  the  value  of  the  following: 

423                                                   65 

7.    73                             8-  35 

««    812 
10.  ^ 

423     /42y_63              8* 
P  "W/                9-44 

152 

111.  The  power  of  a  power.    The  following  example 
illustrates  how  to  find  the  power  of  a  power: 

(ab)3  =  a5  •  a5  •  a5  =  o16 

Similarly,  show  that  (24)3  =  212;   (a3)2  =  a6 
Show  that  (O  n=amn=  (an) m 

Express  this  law  in  words, 


EXERCISES 

Simplify  the  following: 

1.  [(-2)2]3  4.  (a4)3  7.  (x*-b)°+b 

2.  (-22)3  5.  (62)3  8.  (a*-4)*-1 

3.  (z6)2  6.  (a»+i)2  9.  (a?*)*~2 


108  THIRD-YEAR  MATHEMATICS 

MISCELLANEOUS   EXERCISES 

112.  Write  the  following  in  the  simplest  form: 
.     /2a263\3  /3xy\*     /9x\* 

0    /3a»\»     /36Y  6.   \(_^t)ZV 

/a?b4cn\n 

3#  mi5  7-  vavv 

(x62)3  8.  (a2-62)2 


/_2&3x\2"  /a3+63\ 

V    5oV  9#  Va+6/ 


Zero  Exponents.    Fractional  and  Negative  Exponents 

113.  The  symbol  am  is  a  brief  expression  for 

a  •  a  •  a . . . ..  (m  times) . 

Accordingly  m  must  be  a  positive  integer.  In  the  follow- 
ing we  shall  find  a  meaning  for  negative,  fractional,  and 
2ero  exponents. 

114.  Zero  exponents.  The  law  am+an=am~n  has 
been  shown  to  hold  for  positive  integral  values  of  m  and  n, 
with  the  understanding  that  m  is  greater  than  n. 

If  we  assume  the  law  to  hold  also  for  m  =  n,  we  have 

?L  =  a2-2  =  a°;     -  =  a3~s  =  a° 
a2  a3 


However,  by  dividing  numerator  and  denominator  of 
each  fraction  by  the  common  factors  a,  we  have 


a2     <*  •  £       '     a3    #  • 


1 


Thus  we  may  think  of  a0  as  Me  result  obtained  by  divid- 
ing a  power  by  itself,  and  we  may  assign  to  it  the  value  1. 


EXPONENTS.     RADICALS.     IRRATIONALS        109 

In  general,  if  we  assume  the  law  am-i-an  =  am~n  to  hold 
for  m  =  n}  we  have 

CLm 

j    ^-~—  =  am-m  =  a° 
am 

am 
By  reducing  the  fraction  — -  to  the  simplest  form  we 

a 

have  ^=i 

am~ 

Hence  we  may  assign  to  the  symbol  a0  the  value  1,  i.e., 
o°=l 

It  should  be  added  that  a  must  not  be  zero.     For, 

0m    0 
0TOis  0,  and  ^—  =  ^  has  no  meaning. 

EXERCISES 

Give  the  value  of  each  of  the  following  expressions:    10°; 
z°;   (-15)°;   (x'+y)°;   (a-6+c)°. 

115.  Negative  exponent.    Assuming  the  law 


to  hold  also  if  m<  n,  we  have  for  m  =  3  and  n  =  5 : 

"     a3-5  =  a-2 
a5 

a3 
Since  by  reducing  -j  to  the  simplest  form,  we  have 

a3 _</l  ♦  4  ■  ^  _  1 

a5     fi  >  &  >  fi  •  a  •  a     a2 

we  may  de^ne  «~2  to  mean  — 2 . 

Similarly,  or1--,  a~3  =  -3J  a~4  =  -4- 

tt  a  Ui 


In  general,  a  m  = 


110  THIRD-YEAR  MATHEMATICS 

EXERCISES 

Find  the  value  of  each  of  the  following  expressions: 

1.  25  •  2°  4.  8  •  2~4  7.  3-2 

2.  6(a-6)°  5.  53-  5~2  8.  .125"1 

•^       «•©-'      •■©- 

Change  the  following  to  identical  expressions  free  from  nega- 
tive exponents  and  simplify: 


15. 


\3zx-V 


10.  a8  •  a~2 

11.  (-2x-4)(-Sx~1) 

12.  -4a4-a~«  1(J    (z2?/-1)-2 


13.  (2a)~262 


(a26~2)- 


2a-2b  17.  (a+a-^ia-a-1) 

14,  3a36~4  18.  (rc-far"1)2 

Solve  the  following  equations  for  x,  assuming  a^O,  a?*  I: 

19.  a*-2  =  a3  21.  (o»-*)*-W(o«-*)i-* 

20.  a2aj+5=a7-*  22.  4*+*  =  8  •  2X+2 
116.  Fractional  exponent.    Assuming  the  law 

am.an  =  am+n 

to  hold  for  fractional  exponents,  we  have 

a*  •  a*  =  a;  a*  •  a*  •  a$=  a;  a*  •  a*  •  a*  •  a*  =  a;  etc. 

Since  l^a  •.  Va  =  a;  i^a  •.  #"a  •  l^a  =  a; 
Va  •  l/a  •  V a  •  Va  =  a,  etc.,  we  may  define  a*  to  repre- 
sent the  same  number  as  V a.    Similarly, 

a^  =  f/a;  a^  =  Va;  etc. 

i 
In  general,  am=Va 


EXPONENTS.     RADICALS.     IRRATIONALS        111 

/      1\  TO  111 

Show  that  \an)    =  an  •  an  •  an . . . .  (m  factors) 

rrrrr'  "  "  -(wi  terms)  — 

=  a  =a 

1  1 

Show  that  (am)n  =  [a  •  a  •  a. . .  .  (m  factors)]" 
111 
=  an  •  an  •  an . . . .  (m  factors) 

1  /    l\m  m 

.'.  (a")»=UV    =an, 
or  V^=(V~a)m=an 

Thus  a  fractional  exponent  indicates  a  root  of  a  power 
or  a  power  of  a  root,  the  numerator  indicating  the  power 
and  the  denominator  the  root. 


EXERCISES 

Find  the  value  of  each  of  the  following  expressions: 

1.  4*  5.  (-125)*  /27 \-l 

2.  27*  6.  49-*  '  ^8' 

3.  (-8)*  ,27,-*  9.  (-25)* 

4.  (64)*  7#  V64/  10.  (32ar-5&10)* 

117.  Summary  of  the  laws  of  exponents  and  of  the 
meaning  of  fractional,  negative,  and  zero  exponent.     The 

laws  and  definitions  given  in  §§  107  to  116  are  as  follows: 

I.  am>an=am+n  V.  {am)n=amn 

VI.  a°  =  l,  a^O 

a  VII.  <*"m=^ 

III.  {a-b.c)m=ambmcm  i 

VIII.  an=Va 

IV'  \b)   =r  IX.  a*  =  (v/i)m=V/Jl 


112  THIRD- YEAR  MATHEMATICS 

These  laws  have  been  shown  to  hold  for  all  rational 
values  of  m  and  n.  However,  we  shall  find  that,  by  enlar- 
ging our  conception  of  powers,  quite  clear  and  definite 
meanings  can  be  given  to  powers  with  irrational  exponents, 
§  146,  and  even  to  powers  with  imaginary  exponents. 

MISCELLANEOUS  EXERCISES 

118.  Change  the  following  to  identical  expressions 
free  from  negative  or  zero  exponents.  Simplify  as  far 
as  possible: 

2.  \x  V  m  W    •  Viyi 

3.  (VS)-  12.  ^±5^ 

n  —  n — 1 


a— a~ 

4.  \x  *>  )m~n  13.  a-3+b-3 

5.  (a&-2c3)!  a4 


(m-fn\ 
X   *    )' 


14. 


n-i  a-AJra-h 

b  n 
6.  -j-  15.  x-1+2x-2+3x-3 

*  =?  »■  (5SMP~* 

9'  V     a?V     )  18*  U-26/   '  \zb-*J 

Multiply  as  indicated: 

19.  (x*+i/*)(x*— i^)  22.  (ar^+i-V-l+ir,)(jr-1-r"1] 

20.  (z-4+x-2+z)(z-2)  23.  (a*+ai&*+6)(ai-&i) 

21.  (a;-l— 2/~t)3  24.  (ma^+njr^+p)2 


EXPONENTS.     RADICALS.     IRRATIONALS        113 

Divide  as  indicated : 

25.  (a?+a2+ai+a+<z*  +  l)  by  {a+a±) 

26.  (3j/*-f  6x+9atf2/*+2zty)  by  (y+Sx*) 
Extract  the  square  root  of: 

27.  4a2-4a6i+4ac-i+6?-26^c-^+c-: 

28.  x*y-1+4xy-*-2x*-l2xiy*+9y 
Reduce  the  following  fractions: 

x—y         x+y 


29. 


x*+y*  '    &* +2/* 


Exercises  Taken  from  College  Entrance  Examinations 
J119.  Simplify  the  following : 
1.  (2*X2l)-s-54-*     (Sheffield) 


2.  ^(^zf)  +(xUj-s)-l     (Board) 

3.  2(8)1  -  •8(12)1  -2(3)o+  (a*6-»)'6» -g-i+6-i 

4.  (ayx-1)*  -.  (bxy~2)*  •  (?/2a-26-2)i     (Princeton) 
„     (4p4?)3     (p2?5)2 


(Board) 


(9pV)4 '     29 


(Yale) 


°*     ^-2^2   .     a2x-2|/-i     uaie; 

7.  (a4+x4)(a2-z2)-i-(a2-x2)* 

aa^a-^— ax-1) 
x$—a$ 

9.  [^-4a+x-2V6+^6]-[ri-^+2/26]     (Chicago) 

10_  grtfQHtgHl   (M.LT, 

11.  Solve  for  z:   (x+l+x~1)(a;-l+a;-1)  =  5i 


1 14  THIRD-YEAR.  MATHEMATICS 

12.  Multiply:  x-lyi-\-xly~*  by  rri  —  2?/* 

13.  Find  the  square  root  of 
a-2+9^_f_16c-i+6a-i^i_8a-ic-i_2461c-i     (Board) 

X 
■I  _l_0      3 

14.  Find  the  value  of  /o  u,  1flI_2 ,  when  x  =  2     (Cornell) 

15.  Find  the  value  of  x2-\-y2  in  terms  of  u  and  v  when 

,    »-*  u~i 

x  =  u+v — r,   y  =  v+u — r  i 

and  reduce  the  answer  to  its  simplest  form     (Harvard) 

Radicals 
120.    Radical.    Radicand.    Index.     Order.    An   in- 
dicated root  of  a  number,  as  l/5,   #/64,   Va+l,  is  a 
radical. 

The  number  of  which  the  root  is  to  be  taken  is  the 
radicand.  

Name  the  radicands  in  Vs,  2f/x-\-2,  V a. 

The  number  that  indicates  what  root  is  to  be  taken 
is  the  index  of  the  root. 

Give  the  index  in  each  of  the  following  radicals: 
1^26,  Wy,  V%  V~a. 

The  index  gives  the  order  of  the  radical. 

EXERCISES 

1.  Recalling  the  meaning  of  the  square  root  of  a  number, 
find  the  value  of  VW2;  of  V$V$\  of  VaVa;  of  (Va)K 

2.  Find  the  value  of  ^3  ^3^3;  of  flflf  4;  of  fafatfa; 
of  (fa)\ 

3.  What  is  the  meaning  of  Va  f 

n,-         ±  n/-  n/ — 

4.  Since  v a  =  an  show  that  {Va)n  =  a;    v an  =  a. 
6.  Show  that  Va  •  Vb  =  Vab,  or  Va6  =  Va  Vb. 


EXPONENTS.     RADICALS.     IRRATIONALS        115 

6.  State  the  principles  expressed  in  exercises  4  and  5. 

7.  Find  the  value  of  ^9;    ^64;    ^81;    ^32. 

8.  Multiply  {a+Vb)  by  (a-Vb); 

(  Vx+  Vy)  by  Wz-  Vy). 

_    ,  a        x-\-y  m—n 

9.  Reduce  —?=- ;     . ;  -y= — 7= . 

Va     V x+y     V m-\-v  n 

10.  Find  the  value  of  VzVvf;  f9^3;   f~a*Va. 

11.  Find  the  value  of  V 100-36 ;   ^16+9 ; 
(4+3/2)  (4-31^2).    - 

Reduction  of  Radicals 

121.  Removal  of  a  factor  from  the  radicand.    The 

following  examples  illustrate  the  method  of  removing 
factors  from  the  radicand. 

1.  /8=t/T^2=v/4-  i/2  =  2i/2. 

2.  f  16xV=  f  8  •  2  •  x3  •  x2y2  =  ^TTW  2xY  =  2xf  2a; V. 

3.  f  (a+6)2(a2-62)  =  f  (a+6)2(a+&)(a-6)  =  (0+6)  fo^S. 

Thus  a  radical  may  be  reduced  if  the  radicand  con- 
tains factors  to  powers  equal  to,  or  greater  than,  the  index 
of  the  radical. 

EXERCISES 

Reduce  the  following  radicals  to  the  simplest  form: 


1.  VIE 

9.  V21WW& 

2.  V45 

10.  V  a2b2+a?W 

3.  1^98 

11.  ^9a&2-968 

4.  3^40 

12.  i/a3+3a26+3a62+63 

5.   ^1250 

13.  ^Slm^n4 

6.  1^512 

14.  V  S&+18&+27X 

7.  f  8(a+b)6 

8.  5f  24x6^4 

15.  V  (x2-5x+6)(x2-3x+2) 

16.  V a2x4+2abxi+b2x4 

116  THIRD-YEAR  MATHEMATICS 

122.  Reduction  of  a  fractional  radicand  to  the  integral 
form.  The  following  examples  illustrate  the  method  of 
changing  a  fractional  to  an  integral  radicand : 

.181       3/27  ■  3     33/3     3*/3-2s2      3  *,— 

6'  \4z4      \4x3-x     z\4:c     aAUz  •  2z2~2z2     bX 

Hence,  to  change  a  fractional  to  an  integral  radicand, 
reduce  the  radicand  to  the  simplest  form.  Then  multiply 
numerator  and  denominator  by  a  number  which  will  make 
the  denominator  a  power  whose  exponent  is  the  same  as  the 
index  of  the  radical 

EXERCISES 

Reduce  the  following  to  the  simplest  form: 


3  /27a  »/~T 

\    ac 


6-  \w 


5tf_  [g+b 

Sa2y  \a-b 


123.  Reducing  the  order  of  a  radical.     Show  from 
the  following  examples  how  to  reduce  the  order  of  a  radical : 

1.  v^8=(23)i  =  2^=V/2 

2.  Vtf  =  x*  =  xl=fHt?  3.  v/64=^26)i  =  2l  =  f4 

EXERCISES 

Reduce  to  radicals  of  lower  order: 

1.  V25  A.'VM  7.  V9 


2.   ^27  5.  T32  8.   V\2baz¥ 


3.  Va2b*  6.  ^49  9.  i/27x*y3 


EXPONENTS.     RADICALS.     IRRATIONALS        117 

Addition  and  Subtraction  of  Radicals 

124.  Similar  radicals.  If  radicals  when  reduced 
to  the  simplest  form  have  the  same  index  and  the  same 
radicand,  they  are  similar. 

Show  that  3V^6  and  5^6  are  similar  radicals;  also  Vx^y 
and  zVxy. 

If  radicals  are  to  be  added  or  subtracted,  they  are  first 
reduced  to  the  simplest  form.  Similar  radicals  are  then 
combined. 

Thus,  V2+  i/8+3v/50=v/2+2v/2+15i/'2=  18/2 

V  ±x+±y-V  \§x*+l§x2y  =  2V  x+y-±xV  x+y 
=  (2-4x)Vx+y. 

EXERCISES 

Simplify  and  collect  similar  terms: 

1.  4^2-2^54+ #128  „       /3a  ,      /36       \ab 

5*  \y+\  a"\T 


2.  3/98-71/ 80-3/18 


6.  3/ 98-25/3+/108 

7.  28*  +  /63+/ll2 


4 

9.  i/32a"5+i/512a+i/2a  10.  /28-/7+3/175 


11.  4i/l+a2-i/9+9fl2-2i/62+a262 


118  THIRD-YEAR  MATHEMATICS 

Multiplication  of  Radicals 

125.  Multiplication  of  radicals  of  the  same  order. 
Before  multiplying,  all  radicals  should  be  reduced  to  the 
simplest  form  : 

Show  that  1/6  .y 7=  >76^T=  ^42 

3f  xy*c  •  5#ri*=dy&xyc  •  bdfcd2 
=  lhdyf xyc2d2 

EXERCISES 

Multiply  as  indicated: 

1.  Vtf-  Va  4.  (1/2-1/3)1/5 


«2  6.  (2v/5-5)(3-v/5) 


3.  3>^5  •  /10  •  7V^5  6.  (/6+ vlo) (/3 - -/5) 

126.  Multiplication  of  radicals  of  different  orders. 

If  the  radicals  to  be  multiplied  do  not  have  the  same  index, 
they  should  be  changed  to  the  same  order  before  multiplying. 

For  example,  ^Vl  =  ^  .  5*  =  4§  -  &m  V&V&=  ^2000 

Similarly,         4 /sit/  •  Sf^y  =  4:VxY  •  3v^V  =  12v^V 

=  12xVxyb 

EXERCISES 

Multiply  as  indicated: 

1.  V&Vx  3.  V2xf3x2  5.  f2*V\ 

2.  f&Vx*  4.  fa^i/a;  6.  ^9^  •  Vlbx 

Division  of  Radicals 

127.  Division  by  a  monomial.  As  in  multiplication, 
radicals  are  to  be  brought  to  the  same  index  before  divid- 
ing. 

Thus:  V&+  Vxb=  V&+&Vx4V&+&V&*-^yi 

x 


EXPONENTS. 

RADICALS. 

IRRATIONALS 

EXERCISES 

Divide  as 

indicated : 

i/T2 

Vxy 

„    %\rx 

l-w 

3.  —7=- 
Vx 

6.  57= 
2fx 

,  Vl 

2f  54 

fs 

'■71 

4-     71 

6-  v^ 
l/9 

119 


MISCELLANEOUS   EXERCISES 

128.  Simplify  the  following : 


32c~2    96 
S4b      24a  *  8c4 


1    ^ 
*"    >346 

3.  **?(£$*(£)'*     (Sheffield) 

4.  ffcr-S-V'iP;   (\-x)  +  (l-i/x); 

5.  (^)  -^3  •  27-i+(-243)=+(v-^rj      (Chicago) 

6.  -x2(9-x2)~i  +  >/9:^2H — .     3  (Sheffield) 

__   _      V*-(!)2 
7-  VS5-VS+ A17^  (SheffieId) 

8.  /  a3  -  a*b  -  V  a62  -  6s  -  V  (a+b)  (a?  -  b2)     (Chicago) 

9.  3^+1/40+^?--^==     (Sheffield) 

10.  3>I+2"\£_-4aJJ    (Sheffield) 


120  THIRD-YEAR  MATHEMATICS 

11.  /2X^3;  2v/20-i/80+\/^    (Sheffield) 

12.  A^+/63+5/7;    (4v/7-8v/2l+6^42)^-2v/7 

14.  Determine,  without  extracting  roots,  which  one  of  the 
following  is  the  greatest :   t^  10,  /  6 ,  / 17 . 

Rationalizing  the  Denominator 

129.  Rationalizing  the  denominator.  The  process  of 
changing  a  fraction  with  irrational  denominator  to  an 
equivalent  fraction  with  rational  denominator  is  called 
rationalizing  the  denominator.  By  means  of  this  process 
it  is  possible  to  avoid  dividing  by  a  decimal  fraction  when 
the  value  of  the  fraction  is  required. 

1 . 

For  example,  to  find  the  value  of  2  ,  ^*  it  would  be  neces- 
sary to  approximate  the  square  root  of  3,  to  add  the  result  to  2, 
and  to  divide  the  sum  into  1. 

However,  multiplying  numerator  and  denominator  by 
2-i/3,  we  have 

1      _       1(2-/3)        _2-v/3_^2     Vz 
2+/3     (2+/3)(2-/3)       4-3 

Hence  the  value  of y±  may  be  found  easily  by  subtract- 

ing  /3  from  2. 

Moreover,  by  rationalizing  the  denominator  it  is 
possible  to  reduce  the  number  of  square  roots  required  to 
find  the  value  of  a  fraction. 

,     1/5-/2  (/5-Z2)2  7-2/10 

For  example,  y^r2=  {Vb^V-2){Vl-V%— T" 


EXPONENTS.     RADICALS.     IRRATIONALS        121 

It  is  seen  that  the  given  fraction  calls  for  the  approxi- 
mation of  two  square  roots  and  division  by  a  decimal 
fraction.  After  rationalizing  the  denominator  it  is  neces- 
sary to  extract  only  one  square  root  and  to  divide  by  3. 

The  number  by  which  numerator  and  denominator 
are  multiplied  to  make  the  denominator  rational  is  called 
the  rationalizing  factor  of  the  denominator. 

Radical  expressions  of  the  forms  a-\-Vb  and  a—Vb, 
Vx-\-Vy  and  Vx—Vy,  are  called  conjugate  radicals. 


EXERCISES 

Change  the  following  fractions  to  equivalent  fractions  with 
rational  denominator: 

„    VI  z-Vl 

1.    -7=  7.   7= 

Vz  3+1/5 

Vl=VlVz=Vib  3^5-4 

VS     1/31/3"    3  •  2l/5+3 

2    4  9     ^+2/5 

1/2  2/2-3V5 

_    3+/18  31/3-1/7 

3*    vz 


Va+b 

V2 
i/3-l 

V2 


2-V2  3+i/2- 

14.  Find  the  value  of  each  of  the  fractions  in  exercises  6  to  10 
to  three  significant  figures. 


122  THIRD-YEAR  MATHEMATICS 

MISCELLANEOUS  EXERCISES 

|130.  Solve  the  following  exercises: 


1    a  i      *  Vx+2a—Vx-2a     x      _       .. 

1.  Solve  for  x:     ,  -==—     (Board) 

Vx+2a+Vx-2a    2a 

/i/3-V2\2     /V3+i/2\2 

2.  Snnphfy:    (^^(_±_)      (Board) 

3.  Simplify:      P/20+>/12     (Board) 


4.  Solve: 


1/5-1/3 
1  1 


,  1+3^2    3s2-2:r—3     __  .  - 

+T^7i=~^^~    (Yale) 


5.  .bind   the  approximate  value  of  —7= r=-. 7=. 

V2-V6       Vz-2 
to  three  decimal  places.     (M.I.T.) 

6.  Rationalize  and  find  correct  to  two  decimal  places: 

}-'     ,-     (Yale) 

2+i/5-i/2 

1/2+21/3 

7.  Simplify  -7= ^=  and  compute  the  value  of  the  fraction 

to  two  decimal  places.     (Yale) 

8.  Find  the  value  of  x  from  the  equation  5x  =  i/3(l+2x)  and 
express  it  as  a  fraction  having  a  rational  denominator. 

9.  Simplify:  2z2i/9x2+81+27i/4:r.2+36 

Square  Root  of  a  Radical  Expression 

131.  By  squaring  the  binomial  V a-\-Vb  we  have 
a+6+2V/a&.  Therefore  Va-\-Vb  is  the  square  root  of 
the  binomial  (a+6)+2l^a6. 

Hence  it  is  possible  to  find  the  square  root  of  a  binomial 
of  the  form  x+al/y  if  it  can  be  changed  to  the  form 
a+b+2Vab. 


EXPONENTS.     RADICALS.     IRRATIONALS        123 

The  following  examples  illustrate  the  process: 

1.  Find  the  square  root  of  84-1/48. 

8+/48  =  8+2/l2  =  8+2/o^  =  6+2+2/(T~2 
.'.  V/8+/48=/6+/2 

2.  Find  the  square  root  of  38+3/32. 

38+3i/32  =  38+i/9  •  32=38+1/9  -8-4 
=  38+2/72 
=  38+2v/36^2  =  36+2+2i/72 

.\  V38+3/ 32  =  6+ 1/2 

EXEKCISES 

Find  the  square  root  of  each  of  exercises  1  to  9: 
1.3-2/2  4.7+4i/3  7.3-1/5 

2.  6-2/8  5.  11-3/8  8.  7+/I8 

3.  11-4/7  6.  14+6/5  9.  11-6/2 

10.  In  finding  the  sine  of  15°  by  two  different  methods,  we 

obtain  the  results  |(/6— /2)  and  |/2— /3,  respectively. 
Show  that  these  results  have  the  same  value. 

Irrational  Equations 

132.  Irrational  equations  in  the  form  of  quadratics. 

By  changing  the  form  some  irrational  equations  may  be 
solved  like  quadratics. 

For  example,  the  equation 

7x2-5x+l-8/7x2-5x+l=-15 
may  be  changed  to 

a2-8a+15  =  0, 
where  a=  /7x2— 5z+l 

Show  that  ai  =  5,  a2  =  3 

/.  /7x2-5x+l  =  5,  /7x2-5x+l  =  3 
By  squaring  both  sides  of  these  equations  quadratic  equa- 
tions are  found  which  may  be  solved  for  x. 


124  THIRD-YEAR  MATHEMATICS 

EXERCISES 
Solve  the  following  equations: 

1.  a;2-5x+2v/x2-5x-2=10 
Subtract  2  from  both  sides  of  the  equation. 

2.  6y2-3y-2=V2y2-y 
Notice  that  6y2-3y  =  S(2y2-y) 


3.  x2-3x+4+Vz2-3a;+15=19 

4.  2/!-|/§-l  =  0 

5.  4x-^-3x-^-l  =  0 

6.  z-i-5:r-f+4  =  0 

7.  3jrl+20fri-32*0 

8.  (a+2)*-(a+2)i-2  =  0 

9.  f  7^6+4  =  4 i7 7^6 

133.  Irrational  equations  solved  by  reducing  them  to 
rational  equations.  The  following  examples  show  the 
method  of  solving  irrational  equations  which  can  be 
reduced  to  rational  equations. 

1.  Solve  i/z+13-  t/x+6  =  1,  and  check. 

Adding  Vx~+§  to  both  sides,  Vx+\Z  =  l+l/z+6 
Squaring,  3+13  - 1  +2Vx~+§ +x +6 

.'.  3  =  l/^+6 
Squaring  again,  9  =  x +6 

x  =  3 


2.  Solve  t/x+i'2x+1-v/5x+5  =  0 
Isolating  V 5x+5,     Vx-\-V2x+l  =  V5x+5 
Squaring,       x +2'V2xJ+x +2x + 1  =  5x +5 
2V2x2+x=2x+4: 
Dividing  by  2,  V2x2+x  =  x+2 

Squaring,  2x2  +x  =  x2 +4x +4 

.*.  xi  =  4,  x2=-l 


EXPONENTS.     RADICALS.     IRRATIONALS        125 

The  value  x  =  —  1  does  not  satisfy  the  original  equation, 
but  it  satisfies  the  third  equation.  Hence  it  was  brought 
in  by  the  process  of  squaring  the  second  equation.  It 
is  said  to  be  an  extraneous  root  of  the  original  equation. 

Example  2  shows  that  the  results  obtained  by  solving 
an  irrational  equation  are  not  always  roots.  Hence  it  is 
necessary  to  check  all  results  in  the  original  equation. 

EXERCISES 

Solve  the  following  equations  and  check  the  results: 


1.  y+2Vy^l-4:  =  0    (Sheffield) 

2.  V  x~+i+V  2x^1  =  6    (Sheffield) 

3.  i/7x+l->/3x+10=l     (Board) 

4.  V  x+2Q-V~x~=i  =  Z    (Board) 

6.  V2x+§-Vx~=i=Vx~+l     (Princeton) 

6.  V  a-x+VoT+x  =  V  2a+2b 

7.  Vz+5+  v/2x+8=v/7x+21     (Princeton) 

8.  Vlx-b+  VAx- l  =  V 7x-4+V/4x^2     (Harvard) 
Vbx-±+Vb=x    2/x+l 

Vbx-±-  v/5-x~2i/x-l 
Apply  the  process  of  addition  and  subtraction. 


2x-2 


10.  V8x-7 —  =V2x+3 

V2x+3 

Clear  of  fractions. 

11.  V3+x+Vx=   /J— 

^3+s 

J12.  ^4=2V^P2-1  13.  ^L_4  =  2^^i 

Reduce  the  first  fraction  to  the  simplest  form. 

•H*     x~b       ^x-Vb  ,  nWT  a-x    ,    x-6        / r 

Vx+Vb       6  V  a-x     Vx-b 


126  THIRD-YEAR  MATHEMATICS 

J16.  Reduce  V (x-4)2+2/2+  V  (z+4)2+?/2=  10  to  an  equation 
free  from  radicals  and  as  compact  as  possible.     (Board) 

J 17.  Simplify  the  following  expression  as  far  as  possible: 

Yx^+axt-tfx-a?-  V^-3a^+Sa2x-a9-a^4x-4a 

Assume  that  both  x—a  and  x+a  are  positive.  (Harvard) 

Trigonometric  Equations 

134.  Some  trigonometric  equations  reduce  to  irrational 
equations. 

For  example,    tan  0+sec  0  =  3. 

Since  sec  0  =  i/l+tan2  6,  we  have 

i/l+tan2<9  =  3-tan0 

Squaring  both  sides,  1+tan2  0  =  9-6  tan  0+tan2  0 

.-.  6  tan  0  =  8 

4 
tan  0=« 

The  value  of  0  may  be  found  from  a  table  of  tangents. 

EXEKCISES 

Solve  the  following  equations: 

1.  2sin0  =  l+cos0  2.  sin  0+cos  0=  V2 

Summary 

135.  The  chapter  has  taught  the  meaning  of  the  fol- 
lowing terms : 

base  index  of  a  root 

exponent  order  of  a  radical 

power  rationalizing  a  number 

radical  irrational  equations 

similar  radicals  zero  exponent 

conjugate  radicals  negative  exponent 

radicand  fractional  exponent 


EXPONENTS.     RADICALS.     IRRATIONALS        127 

136.  The  following  problems  review  the  essentials  of 
the  chapter: 

1.  Give  the  meaning  of  each  of  the  following: 


I. 

am  .  an=am+n 

VI. 

a°=l 

II. 

„m 

VII. 

am 

III. 

IV. 
V. 

(a  .  b  •  c)m=ambmcn 

fa\m    am 
\bj       bm 

(am)n=amn 

VIll. 
IX. 

an  =  Va 

2.  Explain  how  to  solve  the  following: 

1.  To  change  an  expression  containing  negative  or  zero 
exponents  to  an  identical  expression  free  from  negative  or  zero 
exponents. 

2.  To  remove  a  factor  from  a  radicand. 

3.  To  reduce  a  fractional  radicand  to  the  integral  form. 

4.  To  reduce  the  order  of  a  radical. 

5.  To  add  and  subtract  radicals. 

6.  To  multiply  radicals  of  the  same  order,  or  of  different 
orders. 

7.  To  divide  by  a  radical. 

8.  To  rationalize  the  denominator  of  a  fraction. 

9.  To  find  the  square  root  of  a  binomial  of  the  form  x+avy. 

10.  To  solve  irrational  equations. 

11.  To  solve  trigonometric  equations  leading  to  irrational 
equations. 


CHAPTER  VII 


_41 .RL 


Fig.  58 


LOGARITHMS.     SLIDE  RULE 
Labor-saving  Devices 

137.  Precision  of  measurement.  When  we  compare 
a  line-segment  with  a  known  segment  such  as  an  inch  or 
a  centimeter,  we  are  measuring  the  line-segment. 

To  measure  AB,  Fig.  58,  we  may  lay  off  the  distance  AB 
on  squared  paper,  as  A'B' . 

Using  2  cm.  as  a  unit  A  B 

we  find  the  measure  of  A'B' 
to  be  1.76.  The  6  being 
estimated,  the  number  1 .  76 
does  not  mean  that  the 
length  of  AB  is  exactly 
1 .  76,  but  rather  that  it  is 

between  1 .  755  and  1 .  765.  The  result,  1 .  76,  is  said  to 
be  expressed  to  three  significant  figures,  the  precision 
being  indicated  by  the  number  of  figures.  Three-figure 
accuracy  may  be  obtained  with  ordinary  instruments. 
In  surveying,  four-figure  accuracy  is  usually  sufficient, 
but  with  skill  and  good  instruments  five-figure  accuracy 
is  possible. 

138.  Abridged  multiplication.  In  adding,  subtract- 
ing, multiplying,  or  dividing  two  numbers  obtained  by 
measurement  it  is  useless  to  express  the  result  to  greater 
accuracy  than  that  of  the  less  accurate  of  the  original 
numbers.  Much  labor  may  be  avoided  by  omitting  the 
meaningless  figures  in  the  product  or  quotient.    The 

128 


LOGARITHMS.     SLIDE  RULE 


129 


following  example  illustrates  the  process  of  abbreviating 
the  multiplication  of  two  numbers  which  are  known  only 
approximately : 

Find  the  product  of  2.4301  by  7.8043,  to  five  significant 
figures. 

The  work  may  be  arranged  as  follows: 

2.4301X7.8043 


9 

1944 

17010 


72903 

7204 

08 

7 


18965  22943  . 

A  study  of  the  complete  process  of  multiplying  the  two  num- 
bers brings  out  the  following  facts:  Since  in  7.8043  the  last 
figure,  3,  is  uncertain,  the  first  partial  product,  72903,  is  uncer- 
tain. This  may  be  indicated  by  a  line  drawn  under  each  figure. 
Similarly,  the  4  in  the  second  partial  product,  97204,  is  uncertain, 
etc.  Evidently  the  final  product,  18.96522943,  is  accurate 
only  to  five  significant  figures,  the  last  figure,  5,  being  uncertain. 
Hence  we  may  omit  in  all  partial  products  the  part  to  the  right 
of  the  vertical  line. 

A  further  simplification  is  obtained  by  writing  the  partial 
products  in  the  reverse  order,  i.e.,  multiply  2.4301  by  7,  then 
2.430  by  8,  2.4  by  4,  and  2  by  3. 

The  work  may  now  be  arranged  in  the  following  form: 

2.4301X7.8043 


17.0107 
1.9440 
96 
6 


18.9649 


130  THIRD-YEAR  MATHEMATICS 


* 


EXERCISES 

1.  Find  by  abridged  multiplication  the  following  products : 

12.13X119.4;   14. 625 X  .32814;    .1342X2.16 

2.  A  cubic  centimeter  of  mercury  weighs  13.596  g.,  approxi- 
mately.   What  is  the  weight  of  7 .  43  cubic  centimeters  ? 

3.  How  many  square  inches  are  contained  in  a  square  meter, 
if  a  meter  is  approximately  39 .  37  inches  ? 

139.  Abridged  division.  The  process  is  illustrated 
on  the  following  example : 

Divide  6 .  384  by  1 .  231.  Placing  the  divisor  to  the  right 

6 .  384 1 1 .  231  =  5 .  189  of  the  dividend  and  marking  the 

6  155  uncertain  figures  by  a  line,  we  find 

Z  that  1231  is  contained  in  6384  five 

229  times,  leaving  the  remainder  229. 

123  We  now  cut  off  the  last  figure  in 

the  divisor  and  find  that  123  is 

_  contained  once  in  229,  leaving  the 

_  remainder  106.    The  process  of 

j  q  cutting  off  a  figure  from  the  divisor 

g  is  kept  up  until  the  whole  divisor 

is  used.     Hence  the  quotient  is 

1  5 .  189,  the  9  being  uncertain. 

EXERCISES    k 

1.  Find  the  following  quotients: 

63.4^-26.8;  86.423-5-18.25 

2.  The  lunar  month  has  29.531  days.  How  many  lunar 
months  are  there  in  a  year  which  is  equal  to  365 .  24  days  ? 

140.  Use  of  logarithms.  By  the  use  of  logarithms 
the  operations  of  multiplication  and  division  may  be 
reduced  to  addition  and  subtraction. 

When  only  one  multiplication  or  division  is  to  be  made, 
it  can  be  performed  quickly  by  using  the  abridged  pro- 


LOGARITHMS.     SLIDE  RULE  131 

cesses  given  in  §§  138  and  139.  The  use  of  logarithms 
is  especially  valuable  where  a  series  of  operations  is 
involved. 

The  meaning  of  logarithms  and  the  theory  of  computa- 
tion by  logarithms  will  be  discussed  fully  in  this  chapter, 
§§  145  to  160. 

141.  Use  of  the  slide  rule.  Products,  quotients, 
powers,  and  roots  of  numbers  may  be  found  mechanically 
by  an  instrument  called  the  slide  rule.  A  knowledge  of 
logarithms  is  necessary  to  understand  the  principles  on 
which  the  slide  rule  is  constructed.  A  full  discussion  of 
these  principles  and  of  the  use  of  the  slide  rule  is  found  in 
§§  163  to  166. 

142.  Use  of  tables.  The  student  is  familiar  with 
tables  of  roots  and  powers.  They  may  be  used  to  save 
time  and  to  avoid  unnecessary  labor. 

Logarithms 

143.  Table  of  exponents.  The  table,  Fig.  59,  gives 
the  first  25  powers  of  2.     By  means  of  this  table  it  is 


2  =  2! 

1,024  =  210 

262, 144  =  218 

4  =  22 

2,048  =  2n 

524,288  =  219 

8  =  23 

4,096  =  212 

-    1,048,576  =  220 

16  =  2* 

8,192  =  213 

2,097,152  =  221 

32  =  25 

16,384  =  214 

4,194,304  =  222 

64  =  2* 

32,768  =  215 

8,388,608  =  223 

128  =  27 

65,536  =  216 

16,777,216  =  224 

256  =  28 

131,072  =  217 

33,554,432  =  225 

512  =  29 

Fig.  59 

possible  to  reduce  multiplication  and  division  of  numbers 
to  addition  and  subtraction  respectively.  This  follows 
from  the  theorem  am  •  an  =  am+n. 


132  THIRD-YEAR  MATHEMATICS 

For  example,  let  it  be  required  to  find  the  product 
512X16,384. 

The  table  gives :  512  X  16,384  =  29  X  214  =  223  =  8,388,608. 
Thus  by  adding  the  exponents  9  and  14  we  are  able 
to  locate  the  required  product. 
Similarly,  to  find  the  quotient 

524,288  +  8,192, 
we  find  from  the  table  that 

524,288 -f- 8, 192  =  219-v-213  =  26  =  64. 

EXERCISES 

Find  the  value  of  each  of  the  following: 

1.  16X512  4.  4,194,304+131,072 

2.  2,048X128  5.  8,192-256 

3.  65,536X256  6.  131,072-16,384 

?    128X16X16,384 
*   256X32X17024 

Evidently  a  more  complete  table  of  exponents  would 
be  very  useful  in  performing  multiplications  and  divisions. 

The  following  examples  show  how  the  table  may  be 
used  to  find  powers  and  roots: 

1.  Find  the  square  of  2,048. 
From  the  table  2,048  =  2". 
Therefore  2,0482  =  (211)2  =  222  =  4,194,304. 

2.  Find  the  square  root  of  1,048,576. 
From  the  table  1,048,576  =  220. 

Therefore  1^1,048,576  =  Vlfi  =  (220)  *  =  210  =  1,024. 

The  exponents  in  the  table,  Fig.  59,  are  called  the 
logarithms  of  the  left  members  of  the  corresponding 
equations.     Thus,  if  2  is  used  as  a  base,  the  logarithm 


JOHN  NAPIER 


JOHN  NAPIER,  BARON  OF  MERCHISTON 


JOHN  NAPIER,  a  wealthy  Scotch  baron,  made 
*J  political  and  religious  controversy  the  main 
business  of  his  life,  but  his  pet  amusement  was 
the  study  of  mathematics  and  science.  He  was 
born  in  1550  and  died  in  1617.  The  stupen- 
dous labors  in  calculating  of  some  of  his  contem- 
poraries impressed  him  with  the  desirability  of 
devising  some  way  of  shortening  multiplications 
and  divisions.  Rheticus,  with  forty  helpers,  had 
spent  years  calculating  the  trigonometrical  tables 
published  in  1596  and  1613;  Vieta  seems  to  have 
enjoyed  calculations  requiring  many  days  of  hard 
labor;  Ludolph  von  Ceulen  (1539-1610)  gave 
most  of  his  life  to  calculating  ir  to  35  decimal 
places;  and  Cataldi  (1548-1626)  gave  years  of 
hard  labor  to  numerical  calculating. 

Napier  devised  a  set  of  rods,  known  as 
"Napier's  bones,"  containing  sets  of  products  in 
convenient  form  for  facilitating  multiplying. 

His  virgulae,  another  invention,  were  to  aid  in 
the  extraction  of  square  and  cube  roots.  He  dis- 
covered certain  trigonometrical  formulas,  known 
as  Napier's  analogies,  and  stated  his  "rule  of  cir- 
cular parts,"  a  mnemonic  aid  to  recalling  the  laws 
of  right  spherical  triangles.  His  chief  service  to 
science  was  his  invention  of  logarithms,  which, 
after  suitable  modification  by  Briggs  (1561-1631), 
became  the  powerful  aid  to  calculation  that  we 
employ  today.  His  great  work  on  logarithms, 
entitled  Rabdologia,  was  published  in  1617. 

[See    Ball,    pp.    235-36;    also    Encyclopaedia 
Britannica.] 


LOGARITHMS.     SLIDE  RULE  133 

of  128  is  7.     This  is  expressed  briefly  in  symbols  by  the 
equation 

log2 128  =  7, 

read  the  logarithm  of  128  to  the  base  2  is  7. 

Express  by  means  of  equations  the  logarithm,  to  the  base  2, 
of  the  following  numbers:   16,  256,  2,048,  16,384. 

144.  .Using  3  as  base  we  have  the  following  table  of 
exponents : 

3  =  3*  243  =  35  19,683  =  39 

9  =  32  729  =  36  59,049  =  310 

27  =  33  2,187  =  37  177, 147  =  311 

81  =  34  6,561^=38  531,441  =  312 

EXERCISE 

Express  by  means  of  equations  the  logarithms,  to  the  base  3, 
of  the  following  numbers:  243,  2,187,  19,683. 

145.  Logarithm.*  The  logarithm  of  a  number,  N,  to 
the  base,  a,  is  the  exponent  to  which  a  must  be  raised  to  give 
a  power  that  is  equal  to  N. 

Thus,  if  ax  =  N,  then  logo  N  =  x.  These  two  equations 
are  equivalent. 

*  Tables  of  logarithms  were  first  published  by  John  Napier, 
a  Scotch  baron,  in  1614.  Jost  Biirgi  (1552-1632)  had  calculated 
and  used  extensively  tables  of  logarithms  before  1617,  but  did  not 
publish  his  tables  until  1620. 

Henry  Briggs  (1561-1631)  introduced  the  modern  idea  of 
logarithms  to  the  base  10,  and  it  was  mainly  through  his  influence 
that  logarithms  rapidly  came  into  use  all  over  Europe.  Kepler 
introduced  them  in  Germany  about  1629,  Cavalieri  in  Italy  in  1624, 
and  Edmund  Wingate  in  France  in  1626.  Briggs  also  introduced 
the  method  of  long  division  now  commonly  used  in  arithmetic. 
See  Ball,  Short  Accourit  of  the  History  of  Mathematics,  pp.  235-37,  and 
Tropfke,  Geschichte  der  Elementar-Mathematik,  Band  II,  S.  145-55. 


134 


THIRD-YEAR  MATHEMATICS 


r\ 


EXERCISES 

1.  Using  10  as  base  find  the  logarithms  of  10,  100,  1,000, 
10,000.    Express  each  result  in  the  form  of  an  equation. 

2.  Find  the  logarithm  of  1  to  the  base  1,  2,  3...,  a. 
Express  each  result  in  the  form  of  an  equation.  4*+^  *•  fc 

3.  Find  the  logarithm  to  the  base  2  of  1,  \,  \,  h  TV-  ^  + 

4.  Find  the  logarithm  of  a  number  using  the  same  number  as  J* . 
base. 

Common  Logarithms 

146.  Common  logarithms.  Logarithms  to  the  base 
10  are  called  common  logarithms,  or  Briggs's  logarithms. 
The  base  10  is  generally  not  written,  it  being  understood 
that  10  is  the  base  unless  another  base  is  indicated. 
Thus,  log  x  means  logi0z. 

The  table  of  exponents,  §  150,  contains  only  numbers 
that  are  exact  powers  of  10.  Hence  the  values  of  the 
logarithms  of  these  numbers  can  be  given  exactly.  The 
logarithm  of  a  number  which  is  not  an  exact  power  of  10 
is  written  as  a  decimal  fraction. 

Thus  the  logarithm  of  56.23,  to  five  decimal  places, 
is  1.74997,  since  56.23  =  101-74997,  approximately. 

147.  Characteristic.  Mantissa.  The  integral  part  of 
a  logarithm  is  the  characteristic,  the  fractional  part  the 
mantissa  of  the  logarithm. 

148.  Graphical  representation  of  the  logarithmic  func- 
tion. We  may  use  the  exponential  equation  10y  =  x  to  find 
corresponding  values  of  x  and  y,  satisfying  the  equation 
y  =  log  x.    The  table  below  gives  some  of  these  values. 


y 

0 

1 

1 

1. 

i 

4 

3 

4 

i 

8 

X 

1/10=3.16 

>/l0= 1/3716  =  1.78 

1.783  =  5.62 

1.33 

y 

1 

i 
—1 

4 

l 

~8 

3 

"~  4 

X 

10 

ttot316 

rW=-562 

.749 

.178 

LOGARITHMS.     SLIDE  RULE 


135 


Plotting  these  pairs  of  values  we  obtain  the  graph  of 
the  equation  y  =  log  x,  Fig.  60.     By  means  of  this  graph 


o 
-.1 
— .1 

—.3 

-a 

-.5 


1 


i 


i 


% 


Fig.  60 


it  is  possible  to  find  the  approximate  value  of  the  logarithm 
of  a  given  number. 


136  THIRD-YEAR  MATHEMATICS 

From  the  graph  find  the  logarithm  of  each  of  the 
following  numbers :   §,1.5,4,5,7.4,8.8. 
A  study  of  the  graph  shows  the  following: 

1.  Log  x  is  negative  for  x<l,  and  increases  numeri- 
cally without  bound  as  x  approaches  0. 

2.  As  x  increases  indefinitely,  log  x  also  increases 
without  bound. 

3.  There  are  no  logarithms  of  negative  numbers. 
Laws  of  physics  and  of  other  sciences  frequently  are 

of  the  form   of  logarithmic   or  exponential  equations. 

Fig.  61  represents  a  weight,  w,  suspended 

by  a  rope  wound  a  number  of  times  about 

a  wooden  beam  and  kept  from  falling  by 

a  tension,  t.    The  relation  between  w  and 

t  is  given  by  the  equation  w  =  temx,  m 

being   a   constant   depending   upon  the 

friction  between  the  beam  and  the  rope,  fig.  61 

x  being  the  number  of  times  the  rope  is 

wound  around  the  beam,  and  e  being  equal  to  2.718, 

approximately. " 

exercises 

1.  Graph*  the  equation  y  =  \og2  x  and  discuss  the  properties 
of  the  function  log2  x. 

2.  Why  can  neither  1  nor  0  be  used  as  base  for  a  system  of 
logarithms  ? 

149.  Table  of  logarithms.  Logarithms  may  be  found 
from  special  tables.  From  the  same  tables  we  can  find 
the  number  corresponding  to  a  given  logarithm.  Only 
mantissas  are  given,  the  characteristic  being  determined 
by  a  simple  rule,  §  150.  If  the  given  quantities  are 
measured  accurately  to  three  significant  figures,  a  four- 
place  table  will  avoid  inaccuracies  which  might  affect 


LOGARITHMS.     SLIDE  RULE  137 

the  results.  A  five-place  table  will  give  results  correct 
to  four  significant  figures,  etc.  If  angles  are  measured 
to  the  nearest  minute  only,  four-place  tables  are  suffi- 
ciently accurate.  If  angles  are  measured  to  the  nearest 
second,  five-place  tables  should  be  used. 

150.  Determination  of  the  characteristic*  The  char- 
acteristic of  the  logarithm  of  a  number  is  determined 
by  a  rule  which  may  be  derived  from  the  following 
table: 


From  105   =  100,000, 

follows  that  log  100,000 

=  5 

FromlO4   =   10,000, 

follows  that  log  10,000 

=  4 

FromlO3   =     1,000, 

follows  that  log     1,000 

=  3 

From  102   =        100, 

follows  that  log        100 

=  2 

From  101   =          10, 

follows  that  log         10 

=  1 

From  10°  =            1, 

follows  that  log            1 

=  0 

FromlO-1**              .1, 

follows  that  log 

1 

=  -1 

From  10-2=              .01, 

follows  that  log 

01 

=  -2 

Froml0-3=              .001 

,  follows  that  log 

001 

L=-3,etc 

Hence  the  characteristic  of  the  logarithm  of  any  number 
between  10,000  and  100,000  is  4;  between  1,000  and 
10,000  is  3;  between  100  and  1,000  is  2;  between  10  and 
100  is  1;  between  0  and  10  is  0;  between  .1  and  0  is 
—  1,  etc. 

*The  name  "logarithm"  is  due  to  Napier  (1614)  (Tropfke, 
Band  II,  S.  176). 

Cotes  (1652-1716),  of  Cambridge,  England,  first  speaks  of  a 
system  of  logarithms  (Tropfke,  Band  II,  S.  177).  Briggs  in  1624 
introduced  the  word  characteristic,  and  Wallis  in  1685  introduced 
the  term  mantissa  in  its  modern  mathematical  sense  (ibid.). 

The  phrase  base  of  a  system  of  logarithms  has  been  current  since 
Euler's  time  (1707-83).  Trigonometries  containing  tables  of 
trigonometric  functions  have  been  published  since  Vieta's  time 
(1540-1603).  See  Cajori's,  or  Ball's,  history  of  mathematics  on 
the  topic. 


138  THIRD-YEAR  MATHEMATICS 

The  table  on  p.  137  shows  the  following: 

The  characteristic  of  a  number  greater  than  1  is  one 

less  than  the  number  of  digits  to  the  left  of  the  decimal  point. 

The  characteristic  of  a  number  less  than  1  is  negative  and 

one  greater  numerically  than  the  number  of  zeros  between  the 

decimal  point  and  the  first  significant  figure  of  the  number. 

EXERCISE 

What  is  the  characteristic  of  each  of  the  following  numbers: 
32;  8;  2,468;    .8;    .0021? 

151.  By  means  of  a  table  of  logarithms  we  find  that 

log  7124  =  3.85272 
It  follows  that  7124  =  103-85272 

Dividing  both  sides  of  this  equation  successively  by 
10,  we  have  the  following  equations : 

712.4         =102-^272    .-.log  712.4         =2.85272 
71 . 24       =  101-85272    .-.  log    71 .  24       =  1 .  85272 
7.124     =10-85272     .".log      7.124     =0.85272 
.7124  =1085272~1  .*.  log        .7124  =0.85272-1 
. 07124  =  10-85272"2  .*.  log        .07124  =  0.85272-2,  etc. 

This  shows  that  a  change  in  the  position  of  the  decimal 
point  of  a  number  changes  the  characteristic  of  the 
logarithm,  but  leaves  the  mantissa  the  same.  Thus  all 
numbers  having  the  same  significant  figures  in  the  same 
order  have  the  same  mantissa. 

It  is  customary  to  replace  the  negative  character- 
istics -1,  -2,  -3,  etc.,  by  9-10,  8-10,  7-10,  etc., 
respectively. 

For  example, 
log  .  7124=  -1+0.85272  is  written  9.85272-10, 
and  log  .07124=  -2+0.85272  is  written  8.85272-10. 


LOGARITHMS.     SLIDE  RULE 


139 


The  Table  of  Logarithms 

152.  The  arrangement  of  a  table  of  logarithms.    The 

general  arrangement  of  a  table  of  logarithms  will  be  under- 
stood from  an  inspection  of  part  of  a  table  given  below: 


N 

0 

l 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

- 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

4 
t).4 

0  8 
1.2 

911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

1 

?« 

912 

929 

^0Q4 

.♦OO^OU 

*019|*023  *02S 

*033  *038 

*042 

3 

■ 

913 

96047 

052 

057 

061 

066 

071 

016 

080 

085 

089 

h 

2.0 

1.6 

914 

095 

o!)9 

104 

109 

114 

118 

123 

128 

133 

137 

& 

2  5 

2.0 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

6 
7 
8 
9 

3 . 0 
3.5 
4.0 
4.5 

2.4 
2.8 
3-.  2 

3.6 



-2 
2 

J 


Fig.  62 

In  the  first  column,  N,  we  find  the  first, three  figures 
of  the  number  whose  logarithm  is  to  be  found.  The 
fourth  figure  of  the  number  is  in  the  fijcgt  line.  The  first 
two  figures  of  the  mantissa  are  in  the  second  columnvthe 
last  three  in  the  column  headed  by  the  fourth  figure  of  the 
number  whose  logarithm  is  required.  An  asterisk  [*] 
before  a  mantissa  indicates  that  the  first  two  figures  of- 
the  mantissa  are  to  be  found  in  the  line  below  the  asterisk. 
If  a  number  has  only  3  figures,  or  less,  the  mantissa  is 
taken  from  the  column  headed  Q.  If  a  number  has  more 
than  four  figures,  the  mantissa  is  found  by  a  process  of 
interpolation  or  by  means  of  the  table  of  proportional 
parts  given  in  the  last  column. 

153.  To  find  the  logarithm  of. a  number.  The  ex- 
amples on  p.  140  illustrate  the  use  of  the  table  in  finding 
logarithms. 


140  THIRD-YEAR  MATHEMATICS 

<{.\*&  *V 

1.  Find  the  logarithm  of  912.3. 
The  characteristic  is  2.     Why  ? 

In  the  column  headed  N  find  the  first  three  digits,  912. 
In  the  same  line  and  in  the  column  headed  3  find  the 

mantissa,  96014,  the  asterisk  indicating  that  the  first 

two  digits  are  96. 

.'.log  912.3  =  2.96014 

2.  Find  the  logarithm  of  9126.3. 
The  characteristic  is  3.     Why  ? 

In  the  column  headed  N  find  the  first  three  digits,  912. 
In  the  same  line  and  in  the  column  headed  6  find  the 
mantissa  96028. 

-       /.log  9126.0  =  3.96028 

Similarly,  log  9127 . 0  =  3 .  96033 

This  shows  that  an  increase  of  1  in  the  number  produces  in 
the  mantissa  an  increase  of  .00005,  i.e.,  an  increase  of  5  in  the 
fifth  decimal  place.  It 
will  be  assumed  that  for 
equal  increases  in  the 
number  the  correspond- 
ing increases  in  the 
logarithm  are  nearly 
equal.  For  the  error 
involved  is  very  small 
for  sufficiently  large 
numbers  and  may  be 
neglected. 

To  show  the  meaning 
of  this  assumption  graphically,  let  OA,  OB,  OD,  etc.,  Fig.  63, 
represent  values  of  x,  and  let  AA",  BB",  DD",  etc.,  represent 
the  corresponding  values  of  log  x.    We  are  assuming  that 
B'B"  =  C'C",  approximately,  if  AB  =  BC. 

Moreover,  for  BD  =  T\  BC  we  assume  D'D"  =  T\  C'C". 
Since  9126.3  is  T\  of  the  difference  between  9126  and  9127,  we 
may  find  its  logarithm  by  adding  y8^  of  the  difference  between 
log  9126  and  log  9127  to  the  logarithm  of  9126. 


LOGARITHMS.     SLIDE  RULE  141 

/.  log  9126.3  =  log  9126+fV  (log  9127-log  9126) 
=  3.96028+TV(.00005) 
=  3.96028+.  000015 
=  3.960295 

Since  all  logarithms  in  the  table  are  given  only  to  five  decimal 
places,  the  5  in  the  sixth  place  should  be  omitted, 
.-.log  9126.3  =  3.96029 

The  tables  of  proportional  parts,  headed  P.P.,  Fig.  62,  greatly 
facilitate  the  preceding  computation.  Thus,  for  a  difference 
of  5  in  the  fifth  places  of  two  successive  logarithms,  we  find 
f^-  of  5=1.5  in  the  third  line  of  the  table  of  proportional  parts 
which  is  headed  5. 

EXERCISES 

1.  Using  the  table,  Fig.  62,  find  the  logarithm  of  each  of  the 
following  numbers:  91.457;  0.91014;  0.0009152. 

2.  Using  a  table  of  logarithms  find  the  logarithm  of  each  of 
the  following  numbers: 

1.  9;  27;  342;  875;  964. 

2.  2,028;  4,516;  9,237. 

3.  75,823;  64,003;  0.85992;  0.0041357. 

154.  To  find  the  number  corresponding  to  a  given 
logarithm. 

Given  log  N  =  8 .  96120-10.    To  find  N. 

The  mantissa  of  log  N  is  found  to  He  between  the  mantissas 
96118  and  96123,  corresponding  to  the  numbers  9145  and  9146, 
respectively. 

Thus,  mantissa  of  log  9145  =  96118 
mantissa  of  log  A7"  =96120 
mantissa  of  log  9146  =  96123 

It  is  seen  that  an  increase  of  5  in  the  mantissa  produces  an 
increase  of  1  in  the  number.  From  the  assumption  that  equal 
increases  in  the  number  produce  equal  increases  in  the  logarithm, 
it  follows  that  an  increase  of  2  in  the  mantissa  produces  an 


142  THIRD-YEAR  MATHEMATICS 

increase  in  the  number  equal  to  f  of  1,  or  .4.  Therefore  the 
figures  of  N  are  91454  and  N  =  0.091454. 

More  conveniently,  the  fifth  digit  of  N  may  be  found  by 
means  of  the  table  of  proportional  parts  as  follows: 

Compute  the  tabular  difference  of  the  mantissas  between 
which  log  N  lies.  This  is  found  to  be  5.  Then  compute  the 
difference  between  the  given  mantissa  of  log  N  and  the  next 
smaller  mantissa  in  the  table.  This  is  found  to  be  2.  In  the 
second  column  of  the  table  of  proportional  parts  headed  5  find 
the  number  nearest  to  2.  The  number  in  the  same  line  and  in 
the  first  column  is  the  fifth  digit  of  N. 

EXERCISE 

Find  the  number  corresponding  to  each  of  the  following 
logarithms:  0.35021;  1.43276;  7.58142-10. 

Properties  of  Logarithms 

155.  The  use  of  logarithms  in  shortening  computation 
depends  upon  several  theorems  which  are  proved  below. 

156.  Logarithm  of  a  product.  Let  M  and  N  be  two 
positive  numbers  whose  product  is  to  be  found. 

Let  loga  M  =  m  and  logo  N  =  n 

then  M  =  amsaidN       =an    Why? 

MN  =  am+n    Why? 
.'.    loga  (MN)=m+n    Why? 
.'.    loga(MAO=logaM+logaAr    Why? 
This  equation  expresses  the  following  theorem:    The 
logarithm  of  a  product  is  equal  to  the  sum  of  the  logarithms 
of  the  factors. 

For  example,  log  (15X27)  =  16g  15-flog  27. 

EXERCISES 

1.  Show  that  log  (MNR)  =log  M+log  A7+log  R. 

2.  Given  log  5=  .6990  and  log  7=  .8451,  find  log  35. 


LOGARITHMS.     SLIDE  RULE  143 

157.  Logarithm  of  a  quotient.    To  find  the  logarithm 

M 

of  the  quotient  -^,  let 

log^  M  =  m  and  loga  N  =  n 

Then  M  =  am  and  N  =  an 

M 

^=am~n    Why? 

•'•     loga  (1rf)=m-n    Why? 


.'.    loga  Ujj  =loga M-log  a  N 

Hence  the  logarithm  of  the  quotient  of  two  numbers  is 
equal  to  the  logarithm  of  the  dividend  minus  the  logarithm 
of  the  divisor. 

For  example,  log  J-  =  log  8  — log  3 

EXERCISE 

Find  log  |;  log  |;  log  *ftK 

158.  Logarithm  of  a  power.  To  find  the  logarithm  of 
M v,  let 

loga  M  =  m 
Then  M  =  am 

Mp=(am)p  =  amP    Why? 
.*.    loga  (Mv)=?np, 
or  loga  (MP)  =p  loga  M 

*    Hence  the  logarithm  of  a  power  is  equal  to  the  exponent 
multiplied  by  the  logarithm  of  the  base. 

EXERCISE 

Show  that  log  (74)  =4  log  7. 

159.  Logarithm  of  a  root.    If  p  =  -  show  that  the 

n 

equation  log0  (ilf  ?)  =  p  loga  M  takes  the  form 

fly—  1 

logalM=^loga  Af, 


144  THIRD-YEAR  MATHEMATICS 

i.e.,  the  logarithm  of  the  nth  root  of  a  number  M  is  equal  to 
the  logarithm  of  M,  divide  by  n. 

J160.  Change  of  base.     Let  p  and  q  be  the  logarithms 
of  N  in  two  systems  to  the  bases  a  and  b,  respectively. 
Then  log0  N  =  p  and  log&  N  =  q 

or  N  =  av       and  N  =  b« 

q  =  logb  N  =  \ogb  (av)  =  p  logj  a 
logb  N  =  \oga  N -\ogb  a 

•  1  M     l°gbN 

Thus,  knowing  the  logarithm  of  N  in  the  system  to 
the  base  6,  we  can  find  the  logarithm  of  N  in  a  system 
to  another  base  a  by  dividing  the  known  logarithm  by 
log6  a. 

EXERCISES 

1.  Find  log3  10;  logs  10. 

2.  Find  the  value  of  log2  ^V-log  ^  ■  10. 

3.  Find  the  value  of  log2  3+log3  2. 


MISCELLANEOUS   EXERCISES 

161.  Find  the  value  of  each  of  the  expressions  below. 
The  outlines  in  the  first  three  exercises  suggest  similar 
arrangements  for  the  other  exercises. 


1.  254X12.26 


log  254  = 
log  12.26  = 


adding,  log  N  ■ 


LOGARITHMS.     SLIDE  RULE  145 


12,483  X.  0452 
8,423 


log  12,483  = 
log  .0452  = 


adding, 


log  8,423  = 


subtracting,  log  N  = 


I.    »|^ 
M  86, 


3|^>/352 
420 


lo    Ar  =  log  49+1  log  352-log  86,420 

log  49  = 
§  log  352  = 


adding, 


log  86,420  = 


subtracting, 

dividing  by  3  log  N 


N  = 


23.40X.8625 
.00459X6.3804  8 


0il 

5.  f  92;   ^183  9.  J 


0.6712 
5.327 


nl/25.7\2  ±      (-2582)2X  (.05805) 

6*  \|V286/  2587X(-316) 

Find  the  numerical   value 

3  0 .  0436  by  logarithms,  then  prefix  the 

\  3 .  187  proper  sign. 

11.  log4  64-log3  9+log2  1  (Yale) 


*  When  a  logarithm  is  to  be  subtracted  from  a  smaller  one,  10 
is  both  added  to  and  subtracted  from  the  minuend.  For  example, 
the  form  of  the  logarithm  2.34778  is  changed  to  12.34778-10. 


146  THIRD-YEAR  MATHEMATICS 

13. 


(v/278.2X2.578)3 
f  .00231X^76.19 


^3.416X^25.9_^Q46     (Board) 


% 


4 

$15.  A  number  N  has  17  significant  figures  to  the  left  of 
the  decimal  point.  What  is  the  characteristic  of  log  Nf  of 
log  (log  N)  ?  How  long  can  this  process  of  finding  successive 
logarithms  be  kept  up  ?     (Harvard.) 

X 16.  Find  by  logarithms  the  first  three  figures  of  the  num- 
ber 261— 1.  How  many  figures  will  this  number  contain? 
(Harvard.) 

17.  Given  log  2  =  0.30103,  log  3  =  0.47712.  Find  log  12; 
log*;  log  I;  log  ^6. 

Exponential  Equations 

162.  Exponential  equations.  Equations  in  which  the 
unknown  occurs  in  the  exponents  are  exponential  equa- 
tions. The  following  example  illustrates  the  method  of 
solving  exponential  equations  by  logarithms: 

Solve  the  equation  5* =354 
Taking  the  logarithm  of  both  members, 
log  5*=*  log  354, 
or  2  log  5  =  log  354 

_ log  354 _ 2. 5490 
*~Tol5~  "0.6990' 

EXERCISES 

Solve  the  following  equations: 

1.  3^  =  226  4.  (3.142)^  =  2.718 

2.  2<  =  437  6.  312~2*  =  243 

3.  10*/  =  2. 71828  6.  7*+3=5 


LOGARITHMS.     SLIDE  RULE 


147 


The  Slide  Rule 

163.  Description  of  the  slide  rule.  The  slide  rule  is 
an  instrument  for  determining  mechanically  products, 
quotients,  powers,  and  roots.  It  consists  of  two  pieces 
of  rule,  Fig.  65,  capable  of  sliding  by  each  other. 

Taking  as  unit  the  length  A-B  on  the  rule,  Fig.  64, 
we  may  mark  off,  beginning  from  one  end,  the  logarithms 


1 


Fig.  64 

of  numbers  from  1  to  10,  or  from  10  to  1,000,  or  from  100 
to  1,000. 
Thus, 
Al  =  log    1=0.  A6  =  log    6=    .78 


A2  =  \og    2  =    .30 

A7  =  log    7=    .85 

A3  =  log    3=    .48 

AS  =  \og    8=    .90 

44  =  log    4=    .60 

A9  =  log    9=    .95 

A5  =  log    5=    .70 

A10  =  log  10  =  1.00     - 

In  general,  the 

logarithm 

of  a  number  is  the  distance 

from  A  to  that  number. 

7     8    9     1 


Mv 


tVt 


Fig.  65 

The  arrangement  of  the  slide  rule,  Fig.  65,  makes  it 
possible  to  find  the  sum  or  difference  of  logarithms  even 
more  rapidly  than  with  the  ordinary  table  of  logarithms. 

For  example,  to  find  the  sum  of  two  logarithms,  as 
log  2+ log  3,  place  scale  B,  Fig.  65,  in  such  a  way  that  the 


148 


THIRD-YEAR  MATHEMATICS 


division  marked  1,  on  scale  B,  falls  directly  under  the 
division  marked  2,  on  scale  A. 

Then  division  3,  on  scale  B,  falls  on  division  6,  on  scale 
A,  and  the  distance  A 6,  or  log  6,  is  equal  to  log  2+ log  3. 

It  is  evident  that  this  process  is  practically  the  same  as 
that  of  finding  the  product  2X3  from  a  table  of  logarithms, 
which  is  as  follows : 

Let  N  =  2  X  3.     Required  to  find  N. 

From  the  table,  log  2  =  0. 3010 
and  log  3  =  0.4771 

adding,  log  N  =  0.7781 

From  the  table,        N  =  6. 

To  find  the  difference  between  two  logarithms,  as 
log  6  —  log  3,  place  division  3,  on  scale  B,  directly  below 
division  6,  on  scale  A.  Then  division  1,  on  scale  B,  falls 
directly  below  division  2,  on  scale  A.  Hence  the  distance 
A  2,  or  log  2,  is  equal  to  log  6  — log  3. 


Compare  this  process  with  that  of  finding  the  quotient 


6 


by  logarithms. 

The  two  preceding  examples  show  how  scales  A  and 
B  may  be  used  to  multiply  and  divide  numbers. 

164.  The  Mannheim  slide  rule.    The  Mannheim  rule, 
Figs.  65  and  66,  has  four  scales,  denoted  A,  B,  C,  and  D. 


•    l:':''i 


Fig.  66 


Scales  C  and  D  are  laid  off  to  a  scale  twice  as  large  as  that 
of  A  and  B.  Hence  the  logarithm  of  a  number  on  scales 
C  or  D  is  represented  by  a  segment  twice  as  large  as  the 


LOGARITHMS.     SLIDE  RULE  149 

segment  representing  the  logarithm  of  the  same  number 
on  scale  A.  For  example,  log  2  on  scale  D  =  2  log  2,  or 
log  4,  on  scale  A;  log  3  on  scale  D  =  log  9  on  scale  A,  etc. 
It  follows  that  a  number  on  scale  A  is  the  square  of  the 
number  vertically  below  on  scale  D,  and  that  a  number 
on  scale  D  is  the  square  root  of  the  number  vertically 
above  on  scale  A.  Therefore  scales  A  and  D  may  be 
used  to  find  the  squares  and  the  square  roots  of  numbers. 
The  Mannheim  rule  gives  results  to  three  significant 
figures  which  is  sufficiently  accurate  for  ordinary  use. 


Fig.  67 

If  greater  accuracy  is  required,  Thacher's  slide  rule,  Fig.  67, 
is  used.  This  rule  gives  results  to  four  or  five  significant 
figures. 

The  distance  from  1  to  2  on  scales  C  and  D,  on  the 
Mannheim  rule,  is  divided  into  10  parts,  and  each  of  these 
is  again  divided  into  10  parts.  These  subdivisions  make 
it  possible  to  read  off  between  1  and  2  numbers  containing 
from  2  to  4  digits,  as  1.2,  1.34,  1.526,  the  6  in  the  last 
number  being  estimated  by  the  eye.  Since  the  mantissas 
are  the  same  for  all  numbers  having  the  same  digits  in 
the  same  order,  we  can  give  the  left  index  the  value  10, 
100,  or  1,000.  Thus,  if  the  value  of  the  initial  1  be  100, 
the  divisions  between  1  and  2  will  be  101,  102  .  .  .  .  110, 
111  ....  120  ....  199;  the  division  between  2  and  3 
will  be  200,  210,  220  ....  ,  etc. 


150 


THIRD-YEAR  MATHEMATICS 


On  account  of  these  subdivisions  scales  C  and  D,  when 
used  to  multiply  and  divide,  give  more  accurate  results 
than  scales  A  and  B. 

165.  The  use  of  the  slide  rule.  The  following 
examples  illustrate  the  use  of  the  rule : 

1.  Multiplication.    To  find  the  product  4X3. 

The  directions  are  given  in  Fig.  68 :  The  index  l  of  scale  C  is  put 
over  one  factor  on  scale  D.     The  product 
is  then  found  on  scale  D  under  the  other 
factor  on  scale  C. 

2.  Division.    To  divide  18  by  3. 

Put  the  divisor  on  C  over  the  divi- 
dend on  D.  Find  the  quotient  on  D  under 
1  on  C,  Fig.  69. 

3.  The  product  of  several  factors, 
several  factors  the  runner,  r,  is  used. 

Follow  the  directions  in  Fig.  70  to  find  the  product  8X6X5X2. 


C  I  Put  r 


D     over  4 


under  3 


find  12  =  4X3 


Fig. 


Put  3 


over  18 


under  1 


find  6  =  18^3 


Fig.  69 
To  find  the  product  of 


c 

Put  1    1  r  to  6  11  to  r 

r  to  5 

1  to  r  1  under  2 

D 

over  8  I 

1  find  480  =  8X6X5X2 

Fig.  70 


4.  Reduction  of  fractions.    To  reduce  - 
Follow  the  directions  given  in  Fig.  71. 


Put  42 


26X16.8X35X18 


42X15X91X1.2 


over  26 


r 

to 
168 

15 
to 
r 

r 

to 
35 

91 
to 
r 

r 

to 
1 

12 

to 
r 

below  18 

find  4  =  result 

Fig.  71 


5.  Squares.     Find  the  value  of  122. 

To  find  the  square  of  a  number  place  the  runner  on  the  num- 
ber on  scale  D,  Fig.  72. 

A  I  Find  144=  122 


The  square  is  found  directly  above, 
on  scale  A. 

*  The  right  index  1  is  used  here, 


1) 


Put  r  on  12 
Fig.  72 


LOGARITHMS.     SLIDE  RULE  151 

6.  Square  root.  To  find  the  square  root  of  a  number  proceed 
as  follows: 

Place  the  runner  on  the  given  number  on  scale  A.  The  square 
root  of  the  number  is  directly  below,  on  scale  D. 

166.  Trigonometrical  computations.  On  the  reverse 
side  of  the  slide  three  scales  are  found.  Scales  S  and  T 
are  the  scales  of  angles.  Scale  A  gives  the  sines  of  the 
angles  in  scale  S,  and  scale  D  the  tangents  of  the  angles  in 
scale  T.  The  third  scale  gives  the  logarithms  of  the  num- 
bers on  scale  D.  By  means  of  these  scales  it  is  possible  to 
find  such  products  as  a  sin  x,  or  a  tan  x. 

The  preceding  rules  exemplify  most  of  the  important 
applications  of  the  slide  rule.  There  are  various  makes  of 
rules,  and  makers  generally  furnish  with  each  rule  a 
pamphlet  giving  complete  instructions  as  to  its  use. 

Summary 

167.  The  chapter  has  taught  the  meaning  of  the  fol- 
lowing terms : 

precision  of  measurement  table  logarithms 

abridged  multiplication  common  logarithms 

abridged  division  characteristic 

logarithm  mantissa 

slide  rule  exponential  equation 

168.  The  following  problems  review  the  essential 
parts  of  the  chapter: 

1.  Explain  the  processes  of  abridged  multiplication  and 
division. 

2.  Discuss  the  uses  of  logarithms  and  the  slide  rule  as  labor- 
saving  devices  in  numerical  calculations. 

3.  Give  a  discussion  of  the  graph  of  the  logarithmic  function. 

4.  State  the  rule  for  determining  the  characteristic  of  a 
logarithm. 


152  THIRD-YEAR  MATHEMATICS 

6.  Explain  (1)  how  to  find  the  logarithm  of  a  number  by 
means  of  the  tables;  (2)  how  to  find  the  number  corresponding 
to  a  given  logarithm. 

6.  State  and  prove  the  theorems  regarding  the  properties  of 
logarithms  used  in  finding  products,  quotients,  powers,  and 
roots. 

7.  Explain  the  use  of  logarithms  in  the  solution  of  expo- 
nential equations. 


CHAPTER  VIII 

LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNCTIONS. 
SOLUTION  OF  TRIANGLES 

Use  of  the  Table  of  Logarithmic  Functions 

169.  Logarithms  of  trigonometric  functions.  In  chap- 
ter vii  logarithms  were  used  to  calculate  expressions  in- 
volving products,  quotients,  powers,  and  roots.  When 
logarithms  are  to  be  used  to  find  the  value  of  an  expression 
involving  trigonometric  functions,  the  values  of  the  func- 
tions could  be  looked  up  in  a  table  of  trigonometric  func- 
tions and  the  logarithms  of  these  values  could  then  be 
found  in  a  table  of  logarithms  of  numbers.  To  save  labor 
the  logarithms  of  the  sines,  cosines,  tangents,  and  cotan- 
gents of  angles  between  0°  and  90°  are  given  in  a  special 
table.  The  logarithms  of  secants  and  cosecants  are  rarely 
used  and  may  be  obtained  from  the  logarithms  of  the 
cosines  and  sines,  respectively. 

170.  Arrangement  of  the  table.  Since  the  values  of 
the  sine,  cosine,  and  tangent  of  angles  between  0°  and  45°, 
and  of  the  cotangent  of  angles  between  45°  and  90°  are 
less  than  1,  their  logarithms  will  have  negative  character- 
istics. To  avoid  negative  characteristics  the  form  9-10, 
8-10,  7-10,  etc.,  is  used  in  place  of  —1,  —  2,  —3,  etc. 
The  — 10  is,  however,  omitted  from  the  table.  Hence,  to 
have  the  true  value  of  the  logarithm,  10  must  be  sub- 
tracted from  the  logarithm  found  in  the  first,  second,  and 
fourth  columns. 

When  the  angle  is  less  than  45°,  the  number  of  degrees 
is  indicated  at  the  top  of  the  page  and  the  number  of 

153 


154  THIRD-YEAR  MATHEMATICS 

minutes  is  given  in  the  left-hand  column.  When  the  angle 
is  more  than  45°  and  less  than  90°,  the  number  of  degrees 
is  indicated  at  the  bottom  of  the  page  and  the  right-hand 
column  gives  the  number  of  minutes. 

171.  To  find  the  value  of  the  logarithm  of  a  function 
of  a  given  angle.  The  following  examples  illustrate  the 
method. 

1.  Find  the  value  of  log  tan  52°50'12". 
The  mantissa  of     log  tan  52°50'  =  12026 
The  mantissa  of     log  tan  52°51'  =  12052 


.-.  The  tabular  difference  for  60"  =        26 
The  difference  for    12"  =  jf  X26  =  5.2 
This  difference  may  be  obtained  quickly  by  means  of  the' 
table  of  proportional  parts  as  follows: 

Changing  12"  to  minutes,  12"=  (^x)  =.2'.    This  means 

that  the  required  number  is  in  the  second  line  of  the  table 
headed  26. 

.'.  log  tan  52°50,12"  =  0.12026+5.2 
=  0.12031 

2.  Find  the  value  of  log  cot  48°25'38". 

log  cot  48°25'  =  9 .  94808-10 

log  cot  48°26,  =  9. 94783-10 
Since   the  tabular   difference  is   equal   to   25,   and  since 

™ )  =  .  63',  we  find  in  the  sixth  line  of  the  table  headed 

25  the  number  15 . 0.  This  means  that  .  6  of  25  is  15.  Similarly 
we  find  that  .03  of  25  is  .75.  Hence,  .63  of  25  is  15.7,  or  16 
units  of  the  fifth-decimal  place. 

Since  the  cosine-function  decreases  as  the  angle  increases, 
we  must  subtract  16  from  the  logarithm  of  cot  48°  25'  to  get  the 
logarithm  of  48°25'38r 

.'.  log  cot  48°25'38"  =  9.94808-10-16 
=  9.94792-10 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        155 
EXERCISES 

Find  the  value  of  the  following  logarithms: 

1.  log  sin  71°23'41"  5.  log  tan  27°25,10" 

2.  log  cos  41°15'35"  6.  log  sin  41°57'36" 

3.  log  tan  39°47'36"  7.  log  tan  37°36.'5 

4.  log  sin  65°58'24"  8.  log  tan  23°13.'3 

172.  To  find  the  angle  corresponding  to  a  given  loga- 
rithmic trigonometric  function.  The  following  examples 
illustrate  the  method. 

1.  Given  log  sin  A  =  9.98357-10.    Find  A. 

We  find  that  the  mantissa  lies  between  the  mantissa  of 
log  sin  74°20'  and  log  sin  74°21',  that  the  tabular  difference  is  3, 
and  that  the  difference  between  the  mantissa  of  the  given  loga- 
rithm and  that  of  log  sin  74°20'  is  1. 

In  the  table  of  proportional  parts  headed  3  we  find  .  9  nearest 
in  value  to  1.  Hence  we  may  write  1=  .9+.1.  In  the  first 
column  and  in  the  same  line  with  .  9  we  find  3. 

Similarly  the  number  nearest  to  .  1  in  the  table  of  propor- 
tional parts  is  .09  and  the  corresponding  number  in  the  first 
column  is  .03. 

.-.  A  =  74°20.'33  =  47°20'21" 

2.  Given  log  cos  A  =  9. 85981  -10.    Find  A. 

The  table  shows  that  the  mantissa  lies  between  the  mantissas 
of  log  cos  43°36'  and  log  cos  43°37: 

The  tabular  difference  is  12. 

The  difference  between  the  mantissa  of  the  given  logarithm 
and  that  of  log  cos  43°36'  is  3. 

Hence  in  the  table  of  proportional  parts  headed  12  in  the 
second  column  we  look  for  the  number  nearest  to  3.  This  is 
either  2 . 4  or  3 . 6. 

Let  3  =  2.4+. 6. 

The  corresponding  numbers  in  the   first   columns  are  2 

and  .05 

.'.  A  =  43°36.'25 


156  THIRD-YEAR  MATHEMATICS 

EXERCISES 

Find  the  value  of  A  in  each  of  the  following  equations: 

1.  log  sin  A  =  9. 97527 -10      4.  log  tan  A  =  0.25936 

2.  log  sin  A  =  8. 73997- 10      6.  log  cos  A  =  9. 94749- 10 

3.  log  cot  A  =  9 .  40146  - 10      6.  log  sin  A  =  9 .  42443  - 10 

Use  of  Logarithms  in  the  Solution  of  Right  Triangles 

173.  Solution  of  triangles.  To  solve  a  triangle  is  to 
find  the  values  of  some  of  the  sides  and  angles  by  means  of 
the  given  sides  and  angles.  In  the  course  of  the  second 
year  right  triangles  were  solved  by  use  of  the  natural  values 
of  the  trigonometric  functions.  We  are  now  able  to  carry 
on  by  logarithms  all  multiplications  and  divisions  involved 
in  the  solution. 

174.  Formulas.  The  relations  between  the  sides  and 
angles  of  a  right  triangle,  Fig.  73,  are  expressed  in  the 
following  formulas: 

B 

a  =  c  sin  A  b  =  c  sin  B 

a  —c  cos  B  b  =  c  cos  A 

a  =b  tan  A  b  =  a  tan  B  ,  / 


a  =b  cot  B  b  =  a  cot  A 

c2  =  a2+b2  Fig.  73 

The  equation  c2  =  a2-\-b2  is  usually  taken  in  the  form 


a  =  i/c2-62  =  l/(c+6)(c-6) 

The  area  of  the  right  triangle  is  given  by  the  formulas 

ao    0    02         c2 

S=2=2V/(c+6)(c-6)  =2^TB  =  2  sin  B  C0S  B 

The  preceding  formulas  are  all  adapted  to  logarithmic 
computation. 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        157 

Taking  the  logarithm  of  both  sides  of  the  equa- 
tions they  take  the  forms:  log  a  =  log  c  +  log  sin  A; 
log  &  =  log  c-flog  sin  B,  etc. 

To  determine  an  angle,  the  tangent  or  cotangent  should 
be  used  because  these  functions  change  more  rapidly  than 
do  the  sine-  and  cosine-functions. 

To  determine  a  side,  it  is  best  to  use  the  sine-  or  the 
cosine-function  of  the  given  angle. 

175.  Arrangement  of  the  solution  of  a  right  triangle. 

The  following  example  illustrates  the  plan  to  be  followed  in 
solving  the  right  triangle: 

Case  I. — Given  the  sides  of  the  right  angle.  To  find  the  angles 
and  the  hypotenuse. 

Let  a  =  418  and  6  =  325,  Fig.  74. 

(a)  Draw  a  figure  marking  the  given  and 
required  parts. 

(6)  The  formulas  to  be  used  in  the  solu- 
tion are:  tan  £  =  -;   c  =  - — 5;  A=90°-£. 

a  sin5 

The  equation  a  =  V  (c+b)(c  —  b)  is  to  be  used  as  a  check. 

(c)  Make  a  detailed  outline  of  the  computation  to  be  made, 

thus: 

log  6=  log  6=  Check:  log(c+&)  = 

log  a=  log  sin  B  =  log  (c—b)  = 


log  tan  B  =                     log  c  = 
B=                           c  = 
A=90°-B  = 

(d)  Carry  out  the  computation 

log  a2  =2  log  0  = 

loga  = 

Compare  this  with 

log  a,  found  above. 

according  to  the  plan  in  (c). 

Case  II. — Given  the  hypotenuse,  c,  and  one  of  the  sides,  b,  of 
the  right  angle. 

The  formulas  to  be  used  are:    sin  B  =  cos  A=-;    a 


c'    w    tan  B' 
a  =  V/(c+&)(c-6). 


158  THIRD-YEAR  MATHEMATICS 

Case  III. — Given  one  angle,  B,  and  one  of  the  sides,  6. 

Use      the       formulas       A=90°-B;      a  =  — K=;       0=-^-^: 

tan  B'  sm  B' 

a  =  V(c+b)(c-b). 

Case  IV. — Given  one  angle,  B,  and  the  hypotenuse,  c. 
Use    the    formulas    A  =90°— B;    b  =  c    sin   B;    a  =  c   cos   B; 
a  =  V(c+b)(c-b). 

EXERCISES 

By  means  of  logarithms  solve  the  following  right  triangles: 

1.  c  =  25,  a  =  22  6.  a=194. 5,  6  =  233.5 

2.  c  =  35 .  145,  A  =  25°24'30"       7.  b  =  547 . 5,  B  =  32°15,24,/ 

3.  a  =  316.5,  c  =  521.2  8.  c  =  672.4,  £  =  35°16'25" 

4.  £  =  23°9',  6  =  75.48  9.  a=     3.414,6  =  2875 

5.  c  =  369.27,  a  =  235.64  10.  a  =  617.57,  c  =  729.59 
Solve  the  following  problems: 

11.  In  order  to  determine  the  width  of  a  river,  a  surveyor 
measured  a  distance  of  100  ft.  between  two  points  A  and  B  on 
one  bank.  A  tree  stood  at  a  point  C  on  the  opposite  bank.  The 
angle  ABC  was  found  to  be  63°40'  and  the  angle  BAC  to  be 
55°35?     Calculate  the  width  of  the  river.     (Yale.) 

12.  The  base  of  a  certain  triangle  is  3,248  ft.,  and  the  base 
angles  are  46°15'[  =  46?25]  and  100°37,[  =  100?62].  Find  the 
altitude. 

Sketch  the  figure  (roughly)  to  scale,  and  see  whether  your 
result  is  reasonable.     (Harvard.) 

13.  A  and  B  are  two  points  on  opposite  banks  of  a  river 
1,000  ft.  apart,  and  P  is  the  top  of  the  mast  of  a  ship  directly 
between  them.  The  angle  of  elevation  of  P  from  A  is  14?33 
(14°20')  and  from  B  the  angle  of  elevation  is  8?17  (8°10').  How 
high  is  the  mast  ?     (Harvard.) 

14.  The  shadow  of  a  tower  standing  on  a  horizontal  plane 
is  observed  to  be  100  ft.  longer  when  the  sun's  altitude  is  30°  than 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        159 

when  the  altitude  is  45?     What  is  the  height  of  the  tower  ?    Do 
not  use  tables,  but  express  the  result  in  terms  of  radicals.     (Yale.) 

15.  A  circle  of  radius  5  subtends  an  angle  of  20°  at  a  point  A, 
and  M  and  N  are  the  points  of  contact  of  tangents  drawn  from  A. 
Find  the  perpendicular  distance  from  M  to  AN.     (Harvard.) 

16.  The  value  of  the  smallest  division  on  the  outer  rim  of  a 
graduated  circle  is  30'[  =  0?50],  and  the  distance  between  the 
successive  graduations,  measured  along  a  chord,  is  0.02  inch. 
What  is  the  radius  of  the  circle  ?     (Harvard.) 

17.  Each  of  two  ships  A  and  B,  415  yd.  apart,  measures  the 
horizontal  angle  subtended  by  a  cliff  and  the  other  ship;  the 
angles  are  48°  17'  and  90°  respectively.  If  the  angle  of  eleva- 
tion of  the  cliff  from  A  is  15°24'  what  is  the  height  of  the  cliff  ? 
(Board.) 

18.  At  the  top  of  an  observation  tower  which  is  200  ft.  high 
and  whose  base  is  at  sea-level  the  angles  of  depression  of  two 
ships  are  observed  to  be  30°32'  and  18°40 .'  At  the  bottom  of  the 
tower  the  angle  subtended  by  the  line  joining  the  two  ships  is 
found  to  be  90?  What  is  the  distance  between  the  ships  to  the 
nearest  foot  ?     (Board.) 

19.  A  man  who  is  walking  on  a  horizontal  plane  toward  a 
tower  observes  that  at  a  certain  point  the  elevation  of  the  top 
of  the  tower  is  10°  and  after  going  50  yd.  nearer  to  the  tower  the 
elevation  is  15?     Find  the  height  of  the  tower.     (Princeton.) 

20.  The  diameter  of  the  moon  is  2,164  mi.  long.  Find  the 
distance  from  the  earth  to  the  moon  if  its  apparent  diameter 
subtends  an  angle  31 '1. 

176.  Isosceles  triangle.  The  perpendicular  from  the 
vertex  to  the  base  divides  the  isosceles  triangle,  Fig.  75, 
into  two  congruent  right  triangles.  Since  a  right  triangle 
is  determined  by  two  parts  it  follows  that  two  independent 
parts  must  be  given  to  solve  the  isosceles  triangle. 


160  THIRD-YEAR  MATHEMATICS 

The  following  equations  are  used 
in  the  solution : 

£+^=90°  ^ 

a  =  2b'  sin  ^  =  26'  cos  £ 


h=  W+l){b'~t)  =l tan  B=h' sin  B 

.  ah 

Area  =  -y 

177.  Regular  polygon.  Lines  drawn  from  the  center 
of  a  regular  polygon  to  the  vertices  divide  the  polygon  into 
congruent  isosceles  triangles.  Denoting  the  side  by  a,  the 
radius  of  the  inscribed  or  circumscribed  circle  by  r,  and 
the  number  of  sides  by  n,  we  have 

-  =  r  sin  -=— ,  or  a  =  2r  sin for  the  inscribed  polygon 

2  2n  n 

and  7%  =  r  tan  -p—,  or  a  =  2r  tan for  the   circumscribed 

2  2n  n 

polygon. 

The  area  in  both  cases  is  one-half  the  perimeter  multi- 
plied by  the  apothem. 

Relations  between  the  Sides  and  Angles  of  Oblique 
Triangles 

178.  By  means  of  certain  relations  between  the  sides 
and  angles  of  oblique  triangles  it  will  be  possible  to  compute 
from  certain  given  parts  the  remaining  parts  of  a  triangle. 
These  relations  are  stated  in  the  form  of  three  laws, 
called  the  law  of  sines,  the  law  of  cosines,  and  the  law  of 
tangents. 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        161 

179.  The  law  of  sines.     1.  Let  h  be  the  length  of  the 
perpendicular  from  C  to  A B  in  triangle  ABC,  Fig.  76. 

Show  that       sin  A  =  T  and  h  =  b  sin  A . 


Show  that       sin  B  =  -  and  h  =  a  sin  B. 
a 

,\  a  sin  B  =  6  sin  A. 

It  follows  that  - — t  =  - — ^  . 
sin  A     sin  B 

By  drawing  a  perpendicular  from 

A  to  BC  we  obtain  in  a  similar  way 


Fig.  76 


sin  B 
a 


sin  C 
b 


sin  A    sin.  B    sin  C 

This  equation  is  known  as  the  law  of  sines.  It  may 
be  expressed  in  words  as  follows:  The  sides  of  a  triangle 
are  proportional  to  the  sines  of  the  opposite  angles. 

In  the  obtuse  triangle  ABC,  Fig.  77, 


sin  A=T  and  h  =  b  sin  A 
o 

sin  x  =  sin  (180  —  5)  =sinB  =  -  and/i 

a 

b  sin  A  =  a  sin  B 

a  b 


a  sin  B 


and 

sin  A     sin  B 

It  will  be  seen  in  §§  182,  187, 

189  how  the  law  of  sines  is  used 

in  the  solution  of  triangles. 


Fig.  77 


180.  Diameter  of  the  circumscribed  circle.    The  con- 

— r  =  — — „  =  — — Ti  has  an  interesting  geo- 
A     sin  B    sin  C  to 


stant  ratio 


sin 


162 


THIRD-YEAR  MATHEMATICS 


metrical  meaning.     If  a  circle  is  circumscribed  about 
AABC,  Fig.  78,  it  follows  that  ZA=  ZD. 

.' .     sin  A  =  sin  D  =  -.,d  denot- 
ing the  diameter. 

d=- — 7. 

sin  A 

Thus  the  constant  ratio  of  the  side  of 
a  triangle  to  the  sine  of  the  opposite 
angle  is  equal  to  the  diameter  of  the 
circumscribed  circle. 


Fig.  78 


181.  The  law  of  cosines.     1.  Let   Z  A,  Fig.  79,  be 
acute. 

Then  a2  =  b2+c2-2cb' 

(The  square  of  the  side  opposite 
the  acute  angle  is  equal  to  the  sum  of 
the  squares  of  the  other  two  sides 
diminished  by  twice  the  product  of  one 
of  those  sides  and  the  projection  of  the 
other  upon  it.) 

Since  b'  =  b  cos  A, 

it  follows  that  a2  =  b2+c2-2bc  cos  A. 

This  means  that  the  square  of  a  side  of  a  triangle  is  equal  to 
the  sum  of  the  squares  of  the  other  two  sides  diminished  by 
twice  the  product  of  these  two  sides 
and   the  cosine   of  the  included 
angle. 

This  theorem  is  the  law  of 
cosines. 

2.  If  Z.A  is  obtuse,  Fig.  80,  fig.  80 

a2  =  b2+c2+2cb' 
Since  b'  =  b  cos  x  =  b  cos  (180  -  B)  =  -  b  cos  B,  it  follows 
that  a2  =  b2+c2-2bc  cos  A. 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        163 

Thus  the  same  equation  holds  for  acute  and  obtuse 
angles  A . 

Similarly,  we  find 

b2  =  c2+a2-2ca  cos  B 

c2  =  a2+fc2-2a&cosC 

Show  that  the  theorem  of  Pythagoras  is  a  special  case 
of  the  law  of  cosines. 

The  following  simple  device  makes  it  unnecessary  to 
memorize  each  of  these  three  equations. 

Imagine  the  letters  a,  b,  c  and  A,  B,  C  placed  on  a  circle, 
Fig.  81.  Following  the  direction  indicated  by  the  arrows 
we  pass  from  A  to  B,  then  to  C, 
and  again  to  A.  By  changing,  in 
this  order,  the  letters  in  the  first 
equation  above,  we  deduce  the 
second  equation. 

Similarly  the  third  equation 
may  be  deduced  from  the  second. 
One  formula  is  said  to  be  obtained 
from  the  other  by  cyclic  substitution.  Fig.  81 

182.  The  laws  of  sines  and  cosines  are  sufficient  to 
solve  oblique  triangles.  For,  if  two  angles  and  one  side 
are  known,  the  equation  A+B+C  =  1&0°  determines  the 
third  angle  and  the  law  of  sines  the  other  two  sides. 

If  two  sides,  a  and  b,  and  the  angle,  A,  opposite  one  of 
them  are  known,  the  third  side,  c,  is  found  by  solving  the 
equation  a2  =  62+c2— 26c  cos  A  for  c.  The  other  angles 
are  then  found  by  means  of  the  law  of  sines. 

If  two  sides  and  the  included  angle  are  known,  the  law 
of  cosines  gives  the  third  side  and  the  law  of  sines  the 
other  angles. 

If  three  sides  are  known,  the  law  of  cosines  gives  the 
angles. 


164  THIRD-YEAR  MATHEMATICS 

However,  the  law  of  cosines  is  not  adapted  to  the  use 
of  logarithms  because  it  involves  terms  and  not  factors. 
The  computation  by  means  of  the  cosine  law  without 
logarithms  is  likely  to  be  tedious  for  numbers  containing 
3  or  4  figures.  Hence  we  shall  now  obtain  formulas  that 
are  adapted  to  logarithmic  computation. 

183.  The  law  of  tangents.  Let  ABC,  Fig.  82,  be  any 
triangle. 


Fig.  82 

With  C  as  center  and  the  shorter  of  the  sides  passing 
through  C  as  radius,  draw  a  circle  cutting  CB  and  AB 
in  D  and  E,  respectively. 

Extend  BC  meeting  the  circle  at  D' 
Draw  CE,  AD,  and  AT>' 

Then  ZD'CA=A+B.     .'.  D'DA  =  ±(A+B)      Why? 
A=ZCEA=ZECB+B  .'.  ZECB  =  A-B    Why? 
.-.     ZDAE  =  ±(A-B) 
ZD'AD  =  90° 
Applying  the  law  of  sines  to  AADB, 

DB    sin  DAB        a-b  =       sinJ(A-£) 
AB~ sin  ADB'0Y     c    "sin  [180°-J(A+£)] 

=  sinf(A-fl) 
"sin|(A+B) 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        165 

„  a— b     sin  %(A—B)  ,. 

HenCe'  -T-  =  Sinl(A+B)  (1) 

Applying  the  law  of  sines  to  AAD'B, 

D'B^sinD'AB 
AB     sin  AD'B ' 

q+b^sin  [D'AD+%(A-B)] 
or     c  sin  AD'B 

=  sin[90°+|(A-£)] 
sin  [90°- J(A+£)] 
=  cos|(A-£) 
cos§(A+#) 

q+6_cos  ^(A— £) 
'  *     ~~cT~  cos  i(A+B) 


(2) 


Dividing  equation  (2)  by  equation  (1), 

q+fr  cosf(A-£) 

c  cos|(A+J5)  a+6    cos£(A-£)    sin  J  (A +5) 

a-b  sin  %(A-B)'  a-b    cos  %(A+B)    sin%(A-B) 

*  sin  $(A+B) 

a-b    tanl(A-B) 

This  is  called  the  law  of  tangents. 

The  formula  may  also  be  written  in  the  form 

b+aJanl(B+A) 
b-a    tan^B-A) 

which  is  to  be  used  if  b>a. 

By  cyclic  substitution  two  similar  formulas  are  ob- 
tained, involving  b  and  c  and  c  and  q,  respectively. 


166  THIRD-YEAR  MATHEMATICS 

Since 

A+B    ISO    C_ftAO    C 

2     "2       2~yU      2' 


tan§(A+5)=cot 


it  follows  that 

C 
2 

.'.  By  substitution, 

IX,  cot  o 

a+6  2 


a-b    tan|(A-J8) 
Solving  for  tan  §(A  — B), 

a  form  of  the  tangent  law  to  be  used  when  a,  b,  and  C  are 

known. 

(j 
184.  Mollweide's  equations.    By  substituting  90°— -x 

A-\-B 
for  — 5 —  in  equations  (1)  and  (2),  §  183, 

a-b_sml(A-B)      a+b^cosl(A-B) 
~c~        cos  \C      '       *  sin-C 

These  formulas  are  called  Mollweide's  equations.*    They 
are  especially  useful  for  checking. 

*  Tropfke  (Band  II,  S.  241-42)  says  these  equations  were  pub- 
lished in  1808  by  the  astronomer  Mollweide  (1774-1825),  and  on 
account  of  their  high  applicability  rapidly  found  wide  acceptance 
under  his  name.  The  naming  is,  however,  false  to  history,  for  at 
least  the  second  one  was  known  a  hundred  years  earlier.  Newton 
proved  it  in  substance  in  his  Arithmetica  universalis  of  1707. 

Both  equations  were  derived  as  independent  trigonometric 
theorems  by  F.  W.  De  Oppel  in  his  Analysis  triangulorum  of  1746. 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        167 


185.  Tangents  of  half  the  angles  of  a  triangle.  Radius 
of  the  inscribed  circle.  Let  0  be  the  point  of  intersection 
of  the  bisectors  of  the  angles  of  a  triangle  and  let  r  be  the 
length  of  the  radius  of  the  inscribed  circle.  Let  x,  y, 
and  z  denote  the  lengths  of  the  tangents  from  A,  B,  and  C, 
respectively. 

Then  the  perimeter  =  2x+2y+2z. 

Denoting  the  perimeter  by  2s  and  dividing  by  2, 
s=x+y+z 
Since,  a=       y-\-z 


Similarly, 


s  —  a  =  x 

s-b  =  y; 

From  A  A  OD,  tan  —  =- 
2     x 


tan 


—  c  =  z 


Fig.  83 


Similarly, 


tan  -=  = 


2    s-b 


.      C 
tan  -=  = 


2    s-c 


From  plane  geometry  it  is  known  that  the  area  F  of 
A  ABC  is  given  by  the  formulas 


F  =  Vs(s-a)(s-b)(s-c) 


Before  Mollweide's  rediscovery  they  are  found  also  ia  Thomas 
Simpson's  Trigonometrie  (1765,  2d  ed.)  and  in  Mauduit's  Principles 
of  Astronomy  of  1765. 

Oppel  derived  the  equations  from  the  law  of  tangents,  Simpson 
gave  a  geometrical  proof,  and  Mauduit  was  content  with  merely 
applying  Napier's  Analogies  to  plane  triangles.  Mollweide  derived 
the  equations  from  the  law  of  sines.  His  real  service  was  to  draw 
effective  attention  to  the  great  usefulness  of  these  equations  in 
astronomy. 


168 


THIRD-YEAR  MATHEMATICS 


and 


F  =  2(a+b+c)=rs 
_T/«(s-a)(8-6)(s-c) 


Hence, 


_'i(s-a)(s-b)(s-c) 


H 


Solution  of  Oblique  Triangles 

186.  An  oblique  triangle  can  be  constructed  if  three  of 
the  six  parts  are  known,  at  least  one  of  these  being  a  side. 
Hence  in  solving  oblique  triangles  we  shall  consider  the 
following  four  cases. 

187.  Case  I.  Given  one  side  and  two  angles.  The 
following  example  illustrates  the  method  of  solution: 


Given. 


Formulas: 


A  =  49°38'30" 
£  =  70°21'15" 
6  =  229.38 

C=  180°-  (A  +  B) 

6  sin  A 
a  = 


Required:  C,  a,  and  c 


c  = 


sin  B 

b  sin  C 

sin  B 

Solution:  180°  =  179°59W 
A+£  =  119°59'45' 


.*.     C  =  60°       15" 
log  6  =  2.36055 
log  sin  A  =9. 88198 -10 

Adding,  12.24253-10 

log  sin  £  =  9.97396-10 

Subtracting, 

log  a  =  2. 26857 
.'.a  =185. 59 


Fig.  84 

log  6  =  2.36055 
log  sin  (7  =  9.93757-10 

Adding,  12.29812-10 

log  sin  5  =  9.97396-10 
Subtracting, 

log  c  =  2. 32416 
.\  c  =  210.94 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        169 

Check:    £-A  =  20°42'45"  log  c  =  2. 32416 

£(£-A)  =  10°21'22"       \ogsm±(B-A)  =9.25473-10 
iC=30°       7"  Adding,  11.57889-10 

6-a  =  43.79  iog  cos  JC  =  9. 93752-10 

log(6-a)  =  1.64137  Subtracting, 

log  (b-a)  =  1.64137 
Since  five-place  tables  are  used,  the  results  are  computed  only 
to  5  significant  figures,  the  last  figure  being  uncertain,  and  angles 
are  taken  only  to  the  nearest  second.     Hence  fractions  of  a 
second  are  discarded. 

188.  Cologarithm.    The  principle  log  -^  =  logl  —  log  N 

may  be  used  to  avoid  the  subtractions  in  the  solution  above. 

Since  log  1  =  0,  we  have  log  t?  =  0— log  N 

1 
=  (10  — log  N)  — 10.     The  logarithm  of  -^  is  called  the 

cologarithm  of  N.  When  more  than  one  addition  and  sub- 
traction is  involved,  the  use  of  the  cologarithm  has  a  real 
advantage,  as  it  is  very  easy  to  subtract  mentally  a  loga- 
rithm from  10.  If  cologarithms  are  used,  the  solution  in 
§  187  is  arranged  as  follows: 

log  b  =  2 .  36055  log  b  =  2 .  36055 

log  sin  A  =9.88198-10  log  sin  C  =  9. 93757 -10 

colog  sin  B  =  0.02604  colog  sin  B  =  0 . 02604 

adding,  log  a  =  2 . 26857  adding,  log  c  =  2. 32416 

EXERCISES 

Solve  the  following  triangles  and  check  the  results: 

1.  a  =  29. 73,  A  =  52°36',  J5  =  67°40' 

2.  a=788,  C=72°12'35",  £  =  55°43'18" 

3.  c  =  3795,  A  =  18°53'22",  £  =  81°12'5" 

4.  6  =  37,  A  =  115°36'24",  £  =  27°18'10" 

5.  c  =  913.45,  A  =  64°56'18",  £  =  47°29'11" 

6.  c  =  327.85,  A  =  40°31'42",  £  =  110°52'54" 


170 


THIRD-YEAR  MATHEMATICS 


189.  Case  II.  Given  two  sides  and  the  angle  opposite 
one  of  them. 

It   is    known   from     c* 
geometry  that  it  is  not 
always   possible  to  con- 
struct a  triangle  with 
these  given  parts. 

1.  We  will  consider  first  the  case  where  the  angle  A 
is  obtuse.  Then  the  side  opposite  A  is  the  greatest  side 
of  the  triangle,  and  one  and  only  one  triangle  can  be  con- 
structed satisfying  the  given  conditions. 

2.  If  Zi  is  a  right  angle,  one  triangle  can  be  con- 
structed. The  solution  of  the  right  triangle  has  been 
discussed  in  §  175. 

3.  If  LA  is  acute,  various  possibilities  may  arise: 
(1)  If  a<h,  the  length  of  the  perpendicular  from  D  to 

AB,  the  circle  will  not  meet  AB,  and  there  is  no  triangle 
satisfying  the  given  conditions,  i.e.,  no  solution  of  the 
problem  exists,  Fig.  85. 


Fig.  85 


(2)  If  a  =  h,  the  circle  will  touch  AB,  and  there  is  one 
solution  of  the  problem,  i.e.,  AADE, 

Fig.  86. 

(3)  If  a>h  and  <b,  the  circle  will 
intersect  AB  in  two  points  F  and  F'. 
There  are  two  solutions,  i.e.,  AADF 
and  AADF',  Fig.  87. 


/  I  k]x 

\ii 

\     /  1 

Vj 

F'"—E- 

•-F      J 

FlG. 

87 

LOGARITHMS.     SOLUTION  OF  TRIANGLES        171 


(4)  If  a  is  equal  to  b,  the  circle  will  meet  A  Bin  A  and  in 
another  point,  F.  There  is  one  solution,  i.e.,  AADF, 
Fig.  88. 


F'^ 


Fig.  88 


Fig.  89 


(5)  If  a>b,  the  circle  will  meet  A B  in  two  points,  F 
and  F',  but  only  AADF  satisfies  the  conditions  of  the 
problem,  Fig.  89. 

Since  the  length  of  the  perpendicular,  h,  may  be 
expressed  trigonometrically  in  terms  of  two  of  the  given 
parts  by  means  of  the  equation  h  =  b  sin  A,  the  preceding 
discussion  may  be  summarized  briefly  as  follows : 

1.  A  >  90° ;  then  a  >  b;  one  solution :  an  obtuse  triangle. 

2.  A  =90°;  one  solution:  a  right  triangle 

3.  A  <90°;  and  if  a<b  sin  A;  no  solution 
if  a  =  b  sin  A;  one  solution 
if                             6  >  a  >  b  sin  A ;  two  solutions 
if                                   a  ^  6;  one  solution 

Ca,se  II  is  called  the  ambiguous  case. 


EXERCISES 

State,  without  solving,  how  many  solutions  are  possible  if 
the  given  parts  are  as  follows: 

1.  A  =  50°42',        a  =  204,  b  =  204 

2.  A  =  20°10.'3,     a =57,    6  =  42 

3.  A  =  74°18,13",a=20,    6  =  75 

4.  A  =  32°6/,         a  =  802,  6  =  785 

6.  A  =  45°,  a=  108,  6=152.71 

6.  A  =  77°17.'6,     a  =  210,  6=196 


172 


THIRD-YEAR  MATHEMATICS 


Fig.  90 


Solve  the  following  triangles: 
7.  a  =  140. 5,6  =  170. 6,A  =  40° 
Discussion: 

log  6  =  2.23198 
log  sin  A  =9. 80807-10  * 

log  6  sin  A  =  1.03005 
log  a  =2. 14768 

.'.  a>b  sin  A  and  there  are  two  solutions,  AABC 

and  AB'C,  Fig.  90. 

n    6  sin  A  c  =  180o_(A+jB)        ^smC 

sin  A 


log  a =2. 14768 
log  sin  C=9.99989-10 
colog  sin  A  =0.19193 

log  c= 2. 33940 

c  =  218.49 
log  a =2. 14768 
log  sin  C'  =  9. 29239-10 
colog  sin  A  =0.19193 

log  AB'  =  1.63200 
c'=A5'  =  42.855 

log  c  =  2. 33940 
log  sin  i(B-A)  =8.99348-10 
colog  cos  £C  =  0.14562 

log  (6 -a)  =1.47850 

log  c'  =  1.63200 
log  sin  i(B'~ A)  =9.84447-10 
colog  cos  \C'= 0.00212 


r  ui  triuiua.    am 

a     '  v~' 

6- 

Solution: 

log  6  = 

log  sin  A  = 

colog  a  - 

csin§  (B—A) 
cos  |  C 

=2.23198 

=9.80807-10 

=7.85232-10 

log  sin  B  i 
.'.  £'=ZA£'C  = 

=9.89237-10 
=51°18.'4 
=  128°41.'6 

B+A: 

C=ZACB-- 

B'+A-- 

;.  C'=Z.ACB'-- 

=  91°18.'4 
=  88°41.'6 
=  168°41.'6 
=  11°18.'4 

Check:    B-A=ll°18.'4 

\{B-A)=  5°39.'2 

JC  =  44°20.'8 

6-a  =  30.1 

log  (6 -a)  =  1.47857 

£'-A=88°41.'6 
J(B'-A)=44°20.'8 
§C'  =  5°39.'2 

8.  a=491.2,    c  =  385.7, 

9.  o  =  629,        c  =  462, 
10.  a  =  723,        c  =  483, 


log  (6 -a)  =  1.47859 


C  =  46°15' 
A  =  46°10' 
A  =  140oll, 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        173 


11.  a  =  342.6,    6  =  745.9,    A=43°35.'6 

12.  a  =  345.46,  6  =  531.75,  .A  =  26°47'32" 

1    13.  In  the  triangle  ABC, 

A  =  37°21',  a  =  93,  6  =  85, 
find  the  angle  B.     (Sheffield.) 

14.  A  road  OA  is  9§  mi.  long  and  makes  an  angle  of  31°  16' 
[  =  31°27]  with  a  straight  beach  OX.  From  the  point  A  two 
straight  roads,  AB  and  AB',  each  6  mi.  long,  run  to  the  beach. 
Find  the  distance  along  the  beach  from  0  to  the  nearer  of  the 
points  B  and  B' .     (Harvard.) 

190.  Case  III.     Given  two  sides  and  the  included 
angle.     The  following  example  illustrates  the  method : 
Given:   £  =  37°33'40",  c  =  95,721,  a  =  25,463 
Required:  A,  C,  and  6. 

c+a 


Formulas:  The  equation  tan  J(C— A)=— —  cot  hB  deter- 


mines J(C— A). 

The  equation  i(C+A)=90°-}£  determines  ±(C+A). 
C  =  |(C+A)+J(C-A) 


6  = 
Solution: 


A  =  i(C+A)-±(C-A) 
csinB   c-a_sin  %(C—A) 


sin  C 


cos  hB 


c-a  =  70,258 
c+a=  121,184 

§£=i8°46'50" 

i(C+A)=71°13'10" 
log  (c- a)  =  4. 84670 
colog  (c+a)  =  4. 91656-10 
log  cot  %B  =  0.46847 


log  c=4. 98100 
log  sin  £  =  9.78505-10 
colog  sin  C= 0.12110 


log  tan  %(P-A)  =  0.23173 

i(C-A)=59°36'30" 

i(C+A)  =  71°13'10" 
C=130°49'40' 
A  =  ll°36'40" 


log  6  =  4.88715 
6=77117 

Check:        log  6=4.88715 
log  sin  J(C-A)  =  9. 93580- 10 
colog  cos  |£= 0.02376 

log  (c-o)  =  4. 84671 


174  THIRD-YEAR  MATHEMATICS 

EXERCISES 

Solve  the  following  triangles  and  check: 

1.  a =748,         6  =  375,         C=63°35'30" 

2.  a  =  486,         6  =  347,         C=51°36' 

3.  a=  34.645,  6  =  22.531,  C  =  43°31' 

4.  a=  145.9,      6  =  39.90,    C  =  92°11'18" 

5.  a  =  540,  6  =  420,  C=52°6' 

6.  a  =  469. 71,    6  =  264.37,    C  =  96°57'48" 

7.  a  =103.21,    6  =  152.37,    C=141°8'54" 

8.  a=  167.38,    6  =  152.37,    C=150°20'6" 
Solve  the  following  problems: 

9.  The  diagonals  of  a  parallelogram  are  83.66  and  92.84 
and  one  of  the  angles  of  their  intersection  is  84° .  28.  Find  the 
sides  and  angles. 

10.  Two  trains  start  from  the  same  station  at  the  same  time, 
one  going  north  at  40  mi.  per  hour,  the  other  going  10°  south  of 
east  at  30  mi.  per  hour.  How  far  apart  will  the  trains  be  at  the 
end  of  three  quarters  of  an  hour  ?     (Harvard.) 

11.  An  aeroplane  is  observed  at  the  same  instant  from  two 
stations  on  a  level  plane,  5,280  ft.  apart.  At  the  first  station  the 
horizontal  angle  between  the  aeroplane  and  the  other  station 
is  18°37'[  =  18?62],  and  the  angle  of  elevation  of  the  aeroplane  is 
37°41'[  =  37?68].  At  the  second  station  the  horizontal  angle 
between  the  first  station  and  the  aeroplane  is  64°16'[  =  64?27]. 
Find  the  height  of  the  aeroplane. 

By  the  horizontal  angle  between  two  points,  A  and  B,  each 
viewed  from  a  point  C,  is  meant  the  angle  between  the  vertical  plane 
through  C  and  A  and  the  vertical  plane  through  C  and  B. 
(Harvard.) 

12.  The  two  diagonals  of  a  parallelogram  are  122  and  44, 
and  they  form  an  angle  of  47°28'[=47?47].  Find  the  lengths 
of  the  sides  and  the  angles  of  the  parallelogram.     (Harvard.) 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        175 


13.  In  a  square  ABCD  a  circular  arc  BD  is  described,  with  A 
as  center  and  AB  as  radius.  If  AB  =  5  ft.,  find  the  distance 
from  the  point  C  to  one  of  the  points  of  trisection  of  the  arc  BD. 
(Harvard.) 

191.  Case  IV.  Given  the  three 
sides.  The  following  example  illus- 
trates the  method: 

Given:      a=116.26,  6  =  172.36, 

c=  149.54,  Fig.  91. 
Required:   A,  B,  and  C  

Formulas:  s=i(.a+b+c),  r=^(fr^Ki=W£z£) 


tan  ^A  = 


s—a 


r 


,  tan  ^B  =  :r!—^  ,  tan  \C 


r 
s—c 


A+B+C  =  180° 
Solution:  2s  =  438. 16 


s  =  219.08 
s-a=102.82 
s-b=  46.72 
s-c=  69.54 


log(s-a)  =  2.01208 

log  (s-6)=  1.66950 

log  (s-c)=  1.84223 

cologs=  7.65940 


10 


Check:      2s  =  438. 16 


logr2=  3.18321 
log  r=  11. 59160- 10 


log  r=ll. 59160-10 
log  (s-a)=  2.01208 

logtan|A=  9.57952-10 
|A  =  20°47'42" 
A  =  61°35'24" 


log  r=  11. 59160- 10 
log(s-6)=   1.66950 

logtanf£  =  9.92210-10 
iB  =  39°53'18" 
£  =  79°56'36" 


log  r=  11. 59160- 10 
log  (s-c)  =   1.84223 

logtanJC=  9.74937-10 
±C  =  29°18'54" 
C=58°37'58" 
Check: 
A+B+C  =m°59'5S" 


176  THIRD-YEAR  MATHEMATICS 

EXERCISES 

Solve  the  following  triangles: 

1.  <z  =  13,         6=14,  c=15 

2.  a  =  77,         6  =  123,         c=130 

3.  a  =  5.953,    6  =  9.639,      c=14.424 

4.  a  =  286,        6  =  321,  c  =  463 

6.  a=12. 653,  6  =  17.213,  c  =  23.106 

6.  a  =  34. 278,  6  =  25.691,  c  =  30.175 

7.  a  =1,017,     6  =  246.3,  c  =  495.98 

8.  a  =  49.17,    6  =  52.82,  c  =  61.34 

Area  of  an  Oblique  Triangle 

192.  There  are  various  formulas  for  finding  the  area 
of  a  triangle.  Some  of  them  are  known  to  the  student 
from  geometry.  In  computing  the  area  of  an  oblique 
triangle,  the  selection  of  the  formula  depends  on  the  given 
parts. 

1.  Given  one  side  and  the  altitude  to  that  side: 
In  this  case 

where  T  denotes  the  area,  6  a  side,  and  h  the  altitude  to 
that  side. 

2.  Given  two  sides  and  the  included  angle: 

Since  T  =  ^bh  and  since  h  =  a  sin  C,  it  follows  that 

T_ab  sin  C 

l~       2 

3.  Given  the  three  sides: 

T=^s(s-a)(s-b)(s-c) 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        177 

4.  Give  one  side  and  two  angles: 

From  the  law  of  sines,  a  =  — — =j- 

sin  B 

Substituting  in  the  equation  T  = —  , 

,  _    b2  sin  A  sin  C 

we  have  T  =  — ^— ; — = — 

2  sin  i? 

5.  Given  the  sides  and  the  radius  of  the  inscribed  circle: 

T=rs 

6.  Given  the  sides  and  the  radius  of  the  circumscribed 
circle: 

We  have  seen  that  -^  =  2fl,  §  180. 
sm  C 

•    n      c 
..     sinC  =  ^ 

Substituting  this  in  the  equation 

T=Ja6sin  C 

gives  T=*ahm-m 


EXERCISES 

Find  the  area  of  each  of  the  following  triangles : 

1.  a=40,         6  =  13,        c=37 

2.  a=10,         6=12,        C=60° 

3.  o=122.5,   c  =  122.5,  £  =  110°31' 

4.  A  =  61°30',  £=44°15',   c  =  163 

5.  a=17,         6  =  113,        c=120 


178  THIRD-YEAR  MATHEMATICS 

Solve  the  following  problems: 

6.  A  regular  pentagon  is  7  in.  on  a  side.  Find  the  area  of  the 
regular  five-pointed  star  obtained  by  extending  the  sides  of  the 
pentagon.     (Harvard.) 

7.  Prove  that  the  area  of  a  parallelogram  is  equal  to  the 
product  of  the  altitudes  divided  by  the  sine  of  an  angle  of  the 
parallelogram.     (Harvard.) 

8.  The  sides  of  a  parallelogram  are,  respectively,  28 .  26  and 
30.15.  The  angle  included  between  them  is  68?29.  Find  the 
area.     (Sheffield.) 

9.  The  side  of  a  rhombus  is  2  in.  and  one  angle  is  65?  Find 
the  area. 

10.  In  a  circle  whose  radius  is  1 1 1 . 3  ft.,  find  the  area  included 
between  a  chord  whose  length  is  129 . 3  ft.  and  a  diameter  parallel 
to  it.     (Board.) 

11.  The  diagonals  of  a  parallelogram  are  12 . 5  ft.  and  12 . 8  ft., 
respectively,  and  their  included  angle  is  52°  16'.  Find  the  sides 
and  the  area  of  the  parallelogram. 

193.  Historical  sketch  of  trigonometry.  The  Rhind  Papyrus 
of  somewhere  from  2000  to  1700  b.c,  sometimes  called  the 
Reckoning  Book  of  Ahmes,  employs  the  term  seqt  as  a  technical 
term  having  the  sense  of  our  word  cosine. 

In  his  Risings  of  the  Stars,  Hypsicles  of  Alexandria  (about 
180  B.C.)  employs  calculatory  processes  with  the  aid  of  chord- 
functions  of  angles.  It  seems  probable  that  he  drew  his  knowl- 
edge of  this  prototype  of  our  method  of  reckoning  with  the 
sine-function  from  the  Babylonians. 

Trigonometry  is  commonly  said  to  have  begun  as  a  branch 
of  astronomy  with  Hipparchus  (b.  about  160  b.c).  Sometime 
before  126  b.c.  he  calculated  for  his  help  on  astronomical  prob- 
lems a  table  of  chords  of  circles  that  was  used  much  as  we  use 
a  table  of  sines  and  for  the  same  purposes. 

Hero  of  Alexandria  (first  century  b.c),  in  his  work  on  cal- 
culating areas  of  regular  polygons,  used  a  multiplier  that  amounts 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        179 

to  using  the  tangent-function,  but  as  he  regarded  the  method  as 
a  part  of  astronomy  and  since  he  was  a  surveyor,  he  gave  the 
method  little  attention.  He  probably  drew  his  suggestion  from 
ancient  Egyptian  sources. 

Menelaus  about  98a.d.  wrote  a  treatise  of  six  books  on 
chords  in  circles  and  made  extensive  use  of  calculatory  processes 
based  on  the  chord-function. 

Ptolemy  of  Alexandria  between  125  and  161  a.d.  wrote  an 
epoch-making  treatise  on  astronomy,  the  Megiste  syntaxis,  in  a 
chapter  of  which  he  revived  Hipparchus'  treatment  of  the  cal- 
culation of  triangles,  including  his  table  of  chords.  Ptolemy 
systematized  the  former  treatment,  added  a  few  minor  improve- 
ments, and  made  considerable  use  of  the  tables  in  calculating, 
much  as  we  use  tables  of  natural  sines.  This  great  work,  which 
stood  for  1,500  years  as  the  unquestioned  authority  on  astron- 
omy, was  translated  into  the  Arabic  tongue  in  the  latter  half  of 
the  ninth  century  under  the  Arabic  title,  the  Almagest.  The 
Almagest  was  accepted  as  the  authority  on  trigonometry  until 
1464  a.d.,  when  Regiomontanus  published  his  De  Triangulis. 

The  Hindus  on  Trigonometry 

Aryabhatta  (b.  476  a.d.)  is  the  earliest  Hindu  scientist  whose 
writings  on  mathematics  have  come  down  to  us.  His  work,  the 
Aryabhattayam,  furnishes  us  an  example  of  a  highly  developed 
trigonometry  employing  technical  terms  for  sine,  cosine,  and 
versed  sine  (  =  1  — cos),  and  containing  a  table  of  sines  of  angles 
for  the  interval  3°45 .' 

Brahmagupta  (b.  598)  published  a  book  entitled  Siddhanta, 
a  part  of  which  was  devoted  to  trigonometry,  and  which,  aside 
from  a  few  improvements  in  problems  over  which  Aryabhatta 
had  blundered,  was  no  advance  upon  the  older  work  of 
Aryabhatta.  He  included  Aryabhatta's  table  of  sines  in  this 
work. 

Bhaskara  (b.  1114  a.d.),  another  Hindu,  calculated  and 
published  a  more  accurate  table  of  sines  for  angular  intervals 
of  V 


180  THIRD-YEAR  MATHEMATICS 

Arabian  Trigonometry 

The  Arabs  were  particularly  devoted  to  medicine  and  astron- 
omy. They  were  not  highly  original,  but  were  noted  borrowers. 
They  exercised  the  best  of  judgment  in  what  they  borrowed, 
drawing  from  Greek  sources  in  the  West  and  from  Hindu  sources 
in  the  East.  They  translated  the  best  from  both  sources  into 
their  own  tongue,  but  met  with  indifferent  success  in  organizing 
it  into  a  unified  body  of  doctrine.  It  was  the  irony  of  fate  that 
they  met  with  little  or  no  success  in  advancing  either  of  their 
pet  sciences,  nor  were  they  highly  successful  even  in  trigo- 
nometry, which  is  closely  allied  to  astronomy. 

Alchwarizmi,  about  820  a.d.,  translated  an  extract  from  the 
Hindu  Siddhanta,  and  called  his  translation  the  Sindhind. 
This  Sindhind  was  essentially  only  the  tables  of  Aryabhatta. 
It  soon  gained  wide  circulation  and  through  it  Arabian  scientists 
became  acquainted  with  the  Hindu  calculatory  processes  of 
trigonometry. 

Al  Battani  of  Damascus  (b.  929  a.d.)  was  the  most  signifi- 
cant Arabian  trigonometrician.  He  improved  the  calculatory 
procedures  of  the  Sindhind  and  published  the  results  of  his  work 
in  a  book  entitled  Stellar  Motions,  which  was  translated  into  Latin 
by  Plato  of  Tivoli  at  the  beginning  of  the  twelfth  century  of  our 
era.  This  translation  became  the  basis  of  the  work  of  Regiomon- 
tanus. 

Abu'l  Wafa  of  Bagdad  (940-998)  made  extensive  use  of  the 
known  trigonometric  functions,  and  also  introduced  the  tangent 
and  cotangent.  He  calculated  tables  of  the  trigonometric  func- 
tions, including  tangents  and  cotangents  for  angles  advancing 
by  15'-intervals. 

More  than  a  century  after  Al  Battani's  book,  a  Western  Arab 
at  Seville  in  Spain,  Dschabir  ibn  Aflah  (between  1140  and  1150) 
was  still  using  the  old  Ptolemaic  chord-functions  in  astronomical 
calculations.  He  seems  to  have  paid  not  the  least  attention  to 
the  more  effective  procedures  of  either  Al  Battani  or  Abu'l  Wafa. 
An  anonymous  work  containing  Dschabir's  methods  became 
widely  disseminated  through  Europe.  It  is  said  that  Copernicus 
made  most  of  the  calculations  on  which  his  conclusions  as  to  the 


LOGARITHMS.     SOLUTION  OF  TRIANGLES        181 

plan  of  the  solar  system  were  based,  after  the  ancient  Ptolemaic 
fashion  of  Dschabir's  book,  and  that  the  great  Copernicus  only 
very  gradually  and  late  in  the  progress  of  his  work  adopted 
Regiomontanus'  new  methods  as  they  had  been  developed  by 
Al  Battani. 

Mediaeval  European  Trigonometry 

Johann  Mueller  (1436-76),  whose  scientific  nom  de  plume 
was  Regiomontanus,  while  a  student  of  mathematics  at  the  Uni- 
versity of  Vienna,  undertook  the  translation  into  Latin  and  an 
analysis  of  the  Almagest,  which  had  probably  been  obtained  from 
the  Moorish  schools  in  Spain.  Regiomontanus,  while  working 
on  the  Almagest,  became  impressed  with  the  importance  of 
getting  from  Byzantium  the  original  Greek  text  from  which 
the  Arabic  translation  had  been  made.  This  he  did,  and  in  his 
De  Triangulis  of  1464  he  gave  the  results  of  his  work  on  the 
trigonometry  of  the  Megiste  syntaxis,  the  Almagest,  the  book  of 
Al  Battani,  together  with  his  own  original  contributions.  The 
De  Triangulis  was  the  first  European  trigonometry  as  such,  and 
it  was  the  book  that  did  as  much  as  any  other  one  influence  to 
bring  about  the  renaissance  of  mathematical  activity  in  Europe. 
Based  on  all  the  significant  work  that  had  previously  been  done, 
De  Triangulis  nevertheless  treats  the  subject  differently  from 
any  of  the  sources  and  in  a  highly  satisfactory  form. 

Other  Europeans  before  Regiomontanus  had  worked  on  one 
phase  or  another  of  trigonometry,  and  indeed  Vieta  (1540-1603) 
had  made  considerable  use  of  the  branch  known  now  as  goni- 
ometry.  Rheticus  (1514-76)  had  even  suggested  the  advisability 
of  considering  the  trigonometric  functions  as  ratios.  But  to 
Regiomontanus  belongs  the  honor  of  a  first  complete  scientific 
treatment  in  the  Western  world. 

Pitiscus  (1561-1613)  was  the  first  to  use  the  word  trigonom- 
etry as  the  title  of  a  book  (in  1595). 

After  Regiomontanus  trigonometry  underwent  several 
minor  improvements  and  extensions  until  under  the  masterful 
hand  of  Euler  (1707-83)  it  assumed  its  final  modern  form. 


182  THIRD-YEAR  MATHEMATICS 

In  conclusion,  it  is  one  of  the  curious  facts  of  history  that 
men  developed  even  to  a  high  state  of  perfection  the  science  of 
spherical  trigonometry  long  before  they  gave  much  attention  to 
plane  trigonometry.  Indeed,  plane  trigonometry  was  developed 
to  supply  a  sound  scientific  basis  for  a  well-wrought-out  spherical 
trigonometry  (Tropfke,  Band  II,  S.  189-200). 

Summary 

194.  The  following  problems  review  the  essential 
parts  of  the  chapter: 

1.  What  is  meant  by  the  logarithms  of  the  trigonometric 
functions  ? 

2.  Explain  how  to  find  the  value  of  the  logarithm  of  a  func- 
tion of  a  given  angle. 

3.  Explain  how  to  find  the  angle  corresponding  to  a  given 
logarithmic  trigonometric  function. 

4.  Discuss  the  use  of  logarithms  in  the  solution  of  right 
triangles. 

6.  State  the  formulas  used  to  solve  right  triangles. 

6.  Explain  how  to  solve  an  isosceles  triangle,  a  regular 
polygon. 

7.  State  and  prove  the  following  laws:  law  of  sines;  law  of 
cosines;   law  of  tangents;   Mollweide's  equations. 

8.  Give  the  formulas  expressing  the  following:  radius  of 
the  circle  circumscribed  about  a  triangle;  radius  of  the  inscribed 
circle;  tangent  of  half  an  angle  of  a  triangle;  area  of  a  triangle. 

9.  Explain  the  meaning  and  the  use  of  cologarithms. 

10.  Discuss  the  cases  occurring  in  the  solution  of  oblique 
triangles  and  state  the  formulas  used  in  each  case. 


CHAPTER  IX 

RELATIONS  BETWEEN  FUNCTIONS  OF 
SEVERAL  ANGLES 

Addition  and  Subtraction  Theorems 

195.  Addition  theorems  for  sine  and  cosine.    The 

discussion  of  these  theorems  is  divided  into  three  steps, 
as  follows: 

I.  Let  a  and  ft  Fig.  92, 
represent  two  positive  acute 
angles  whose  sum  is  less 
than  90° 

Let  ZA£C  =  a+/3 

To  find  sin  (a+0). 

Draw  CD±AB. 

DC 


Then  sin  (a+0)  = 


CB 


Draw  CEl.BE,  EF±AB,  and  EG  LCD 
Show  that 

•    (    .  Q^_DC _DG+GC _FE+GC _FE    GC 
sin  1«H-P;-CB-      C£      -      C£     "CBTCB 


Show  that 
FE  =  EB  sin  a  and  that  7Tn=-?TB  sin  a  =  cos  0  sin  a. 


Since  Z  GCE  =  a,  it  follows  that 

GC  =  CE  cos  a  and  that  ttb=7tb  c°s  a  =  sin  /3  cos  a, 

183 


184  THIRD- YEAR  MATHEMATICS 

By  substitution, 

sin  (a-f- P)  =  sin  a  cos  p+cos  a  sin  p  (1) 

cos  (a+j8)  may  be  found  as  follows:    , 

(       R^BD  =  BF-DF_BF    DF    BF    GE 
cos  (a+p;     CB         CB      -CB     CB-CB    CB 

Show  that 

DP         WD 

BF  =  #£  cos  a  and  that  ~^  =  ttb  cos  a  =  cos  0  cos  a 


Similarly, 

GE  =  CE  sin  a,  and 
By  substitution, 


GE    CE    .  .     .    . 

7rR  =  TrR  Sln  a  =  sin  P  sin  a 


cos  (a+p)  =cos  a  cos  p  —  sin  a  sin  p 


(2) 


Equations  (1)  and  (2)  are  the  addition  theorems  for 
the  sine  and  cosine. 

II.  Equations  (1)  and  (2) 
can  be  proved  for  a+/3  greater 
than  90°,  Fig.  93.  g 

Equation  (1)  may  be 
proved  exactly  as  in  case  I. 

To  prove  equation  (2), 
put 


cos  (a+ff)  =  — 


DB 
CB 


DF-BF 
CB 


,  etc. 


It  follows  that  equations  (1)  and  (2)  hold  for  any  two  acute 
angles. 

III.  Let  one  of  the  angles  in  equations  (1)  and  (2)  be 
increased  by  90? 

Denote  the  sum  by  a',  i.e.,  let  a' =  90° -ha. 


FUNCTIONS  OF  SEVERAL  ANGLES  185 

Then  sin(a'+/3)  =  sin[90o+(a+/3)]  =  cos(a+/3) 
=  cos  a  cos  13— sin  a  sin  0. 

But  cos  a  =  sin  (90° + a)  =  sin  a', 

and  sin  a  =  —  cos  (90°+ a)  =  —  cos  a'. 

By  substitution,  sin  (a'+ /?)  =  sin  a'  cos  /3+cos  a'  sin  /?. 

Thus  equation  (1)  holds  if  one  of  the  angles  is  increased 
by  90? 

Similarly, 

cos  (a'+/3)  =  cos  [90°-+(a-f /?)]  =  -sin  (a+0) 
=  —sin  a  cos  /3—  cos  a  sin  p. 

By  substitution,  cos  (a'+jS)  =  cos  a'  cos  /3— sin  a'  sin  0. 

Thus  equations  (1)  and  (2)  hold  not  only  for  any  two 
acute  angles,  but  they  are  valid  if  one  or  the  other  angle 
be  increased  by  90? 

By  the  same  reasoning  it  follows  that  these  equations 
are  true  if  we  repeatedly  increase  one  or  the  other  angle 
by  90?    Hence  they  hold  for  any  two  angles  whatever. 

EXERCISES 

1.  Compute  sin  75°;  cos  75° 
Put  75°  =  45°+30° 

Then  sin  75°  =  sin  (45°+30°)  =sin  45°  cos  30°+cos  45°  sin 30° 

=  K2-K3+^2.|  =  il±^. 

Compute  the  sine  and  cosine  of  each  of  the  following  angles: 

2.  15°  6.  150° 
Put  15°  =  60° -45?                            6    210° 

3-  90°  7.  195° 

Put  90°  =  60° +30? 

4.  105°  8'  120° 

Put  105°  =  60°  +45?  9.  240° 


186 


THIRD-YEAR  MATHEMATICS 


Verify  the  following: 

10.  sin  (45°+a)  =  (cosa;+sina^ 


11.  cos  (60°+a) 


COS  a 


-Vz 


sin  a 


196.  Subtraction    theorems    for    sine    and    cosine. 

These  theorems  are  proved  as  follows : 
Let  a=(a-/3)  +  0 
Show  that, 
sin  a  =  sin  [(a  —  /5)  +  /3] 

=  sin  (a  —  /3)  cos  /3+cos  (a  —  /3)  sin  /?, 
cos  a  =  'cos  [(a  —  $)  +  /3] 

=  cos  (a  —  0)  cos  /5  — sin  (a  — (3)  sin  /3. 
Denoting  sin  (a  —  /3)  by  z  and  cos  (a  —  /?)  by  ?/,  these 
equations  take  the  form 

sin  a  =  #  cos  &-\-y  sin  /3 
and  cos  a=—x  sin  /3+?/  cos  /3 


Solving  by  determinants, 


sin  a 

sin  /3 

cos  a 

cos  /3 

cos  j3 

sin  /? 

—  sin  /? 

cos  /3 

cos  /? 

-sin  0 


sin  a 
cos  a 


cos  /5 
sin  /? 


sin  /3 
cos  0 


or 
and 


_sin  a  cos  ff— cos  a  sin  ft 
*"•    X~  cos2  0+sin2  0  ' 

cos  a  cos  ff+sin  a  sin  ff 
y"  cos2  0+sin2  fi 

.*.    z  =  sin  a  cos  /3— cos  a  sin  /?, 
2/  =  cos  a  cos  /3+sin  a  sin  /?, 

sin  (a-p)=sinacos  p— cos  a  sin  p 
cos  (a-p)=cosacosp+sina  sin  |5. 


FUNCTIONS  OF  SEVERAL  ANGLES  187 

EXERCISES 

1.  Compute  sin  15°;  cos  15° 

sin  15°  =  sin  (45°-30°)  =sin  45°  cos  30°-cos  45°  sin  30° 
=  ^2.^3-^2.*  =  ^ 

Compute  the  sine  and  cosine  of  the  following  angles: 

2.  0° 
Put0°=30°-30° 

3.  15° 

Put  15°  =  60° -45° 

Apply  the  subtraction  theorems  in  the  following: 

4.  sin  (90°-z)  6.  cos  (30°-*/) 
6.  cos  (45°-z)                               7.  sin  (360° -#) 
Verify  the  following: 

8.  sin  (x-\-y)  sin  (x— y)=sin2  a;— sin2  y 

9.  cos  {x-\-y)  cos  (x— ?/)=cos2  x— sin2  y 

10.  sin  (x+y-\-z)  =sin  x  cos  y  cos  2+cos  x  sin  y  cos  z 

+cos  x  cos  y  sin  0— sin  a;  sin  y  sin  2 

197.  Sums  and  differences  of  sines  and  cosines.    By 

means  of  the  formulas  developed  below,  the  sums  or 
differences  of  sines  and  cosines  can  be  transformed  into 
products. 

sin  (a+/3)  =sin  a  cos  /3+cos  a  sin  (3      (1) 
sin  (a  —  j8)  =  sin  a  cos  /?  —  cos  a  sin  /3      (2) 

Adding  (1)  and  (2), 
sin  (a+/3)+sin  (a  —  j8)=2  sin  a  cos  0  (3) 

Subtracting  (2)  from  (1), 
sin  (a+/3)— sin  (a—  j3)=2  cos  a  sin  /?  (4) 


188  THIRD-YEAR  MATHEMATICS 

Similarly, 
cos  (a+/3)-f-cos  (a  — /3)=2  cos  a  cos  /?  (5) 

and 

cos  (a-f/3)  —  cos  (a  —  (3)  =  —  2  sin  a  sin  (3  (6) 

Let  a+/3  =  A  and  a  —  (3  =  B 

Then  a  =  %(A+B)  and  /?  =  §(A-£)    Why? 

Substituting  these  results  into  equations  (3)  to  (6), 

sin4+sinfl  =  2sin£(i4+5)  cos £(4-5) 
sin4-sin5  =  2cos^CA+£)  sin£(4-fl)" 
cos4+cosJ5  =  2cos*(i4+5)  cos£(i4-fl) 
coSi4-cos5=-2sin£(i4+5)  sin \{A-B) 

EXERCISES 

Verify  the  following: 

1.  sin  35°-J-sin  15° =2  sin  25°  cos  10° 

2.  sin  75°+sin  15°= J^6 

3.  sin  30-hsin  0  =  2  sin  20  cos  0 

4.  sin  5x+sin  3x  =  2  sin  4x  cos  x 

6.  sin  -~— sin  -  =  2  cos  x  sin  - 

6.  sin  (45°+z)+sin  (45°-a;)  =  ^2  cos  2 

sin20+sin0_        fl 
'*  cos  20-cos  6  2 

cos  6x-i-cos  4x 


FUNCTIONS  OF  SEVERAL  ANGLES  189 

198.  Addition  and  subtraction  theorems  for  tangent 
and  cotangent. 

Show  that 

sin  (a+j8)     sin  a  cos  /3+cos  a  sin  (3 


tan  (a+j8)  = 


cos  (a+jS)     cos  a  cos  (3— sin  a  sin  0 

sin  a  cos  ff    cos  a  sin  ff 

_cos  a  cos  /3    cos  a  cos  0 

cos  a  cos  £     sin  a  sin  jS 

t 

tan(a+p) 


cos  a  cos  &     cos  a  cos  (3 
tan  a+tan  p 
1— tana  tan  p 


Similarly, 


tan  a -tan  p 


EXERCISES 

Verify  the  following: 

4        i.  (    \  o\     c°t  «  cot  j8—l 
Lcot(tt+^=cot/8+cot, 

a.  cot(a-^)=cot°cot^+1 

cot  /?  — cot  a 

3.  tan(45+A)  =  1+tanA 


4.  tan  (45- A)  = 


1— tan  A 
1  — tan  A 


1+tan  A 

199.  Functions  of  double  an  angle. 

In  the  equation 

sin  (a+j8)  =sin  a  cos  /3+cos  a  sin  ft 
put  (3  =  a. 

Then      sin  (2a)  =  sin  a  cos  a+cos  a  sin  a, 

or  sin  2a  =  2  sin  a  cos  a. 


190  THIRD-YEAR  MATHEMATICS 

Similarly,  from 

cos  (a+/3)  =  cos  a  cos  0— sin  a  sin  (3, 

it  follows  that 

cos  2a  =  cos2  a  —  sin2  a. 

Since         cos2  a  =  1  —  sin2  a,  we  have 

cos  2a  =  1—2  sin2  a. 
By  means  of  the  equation 

sin2  a  =  1  —  cos2  a 

we  obtain 

cos  2a  =  cos2  a  —  (1  —  cos2  a) 
=  cos2  a  — 1  +  cos2  a 

.*.    cos  2a  =  2  cos2  a  —  1. 

From  the  equation 

tan  a+tan  ft 
tan(a+ft)  =  1_tanatan/3 

we  have 

2  tana 
tan  2a =^ — r — ^— . 
l-tan2a 

EXERCISES 

1.  Show  that  cot  2a =— — - — 

2  cot  a 

»   at.       4.1,  ±  ±      o      3  tan  a -tan3  a 

2.  Show  that  tan  3a =— ; — — — ; 

1—3  tan2  a 

Put  3a=(2o+o). 


FUNCTIONS  OF  SEVERAL  ANGLES  191 


200. 

Functions  of  half  j 

an  angle 

In  the  equation 

cos  2a  =  1  —  2 

sin2  a, 

let 

2a  =  x 

then 

X 

This 

gives  the  equation 

cos  z=l  —  2  sin2  •=, 


x 
Solving  for  sin  =  > 


-^ 


.    JC  1— cos* 

SU1 


2         \       2 

Similarly,  from  the  equation 
cos  2a  =  2  cos2  a  —  1 
we  derive 


cos  ~ 


-V 


2  \       2       • 

x  x 

By  dividing  sin  5  by  cos  5 ,  we  have 


x  1— cos  a 


t 


ll+cosa  SL- 


EXERCISES 

Prove  that  the  following  statements  are  identities: 

.      .    0        2  tan  x 

1.  sin  2x  =  — — 

1+ tan2  re 

sin  2x  =  2  sin  re  cos  re 

sin  re 

2  tan  re  cos  x       2  sin  re  cos  re     2  sin  re  cos  re 


1+ tan2  re  sin2  re     cos2  re + sin2  re  1 

cos2  re 


192  THIRD-YEAR  MATHEMATICS 

2.  (sin— +cos  — )   =l+sinA 

„       sin  2a 

3.  — —  =  tana 

1+cos  2a 

4.  tan  A+cot  A  =  2  esc  2A 

_    tan  z-ftan  y 

6.  — - — ■ — -  =  tan  x  tan  y 

cot  x+cot  y 

«       .  sin  2x 

6.  cot  x 


1— cos  2x 
x 


l-tan22 

7.  » =  cos  x 

1+tan*  | 

8.  1+tan  A  tan  —  =  sec  A 


MISCELLANEOUS   EXERCISES 

201.  Verify  the  following  statements: 
cos  x + sin  x 


1.  tan  2x+sec  2x  = 

2.  2  sin  z+sin  2x  — 


cos  x— sin  x 

2  sin3  x 
1  — cos  x 


3.  (seca+tana)2=^+sma 

1  — sin  a 

4.  sin  (^7r+a)— sin  (^7r— a)=sina 

5.  cos  3z  =  4  cos3  x— 3  cos  z 

6.  1-ftan  a  tan -  =  sec  a 

7.  tan  z  =  tan  -+tan  -  sec  x 

8    cos  (a— 45°)_    cos  2a 
'  cos  (a+45°)~l-sin2a 


FUNCTIONS  OF  SEVERAL  ANGLES  193 

Solve  the  following  problems : 

9.  Given  tan  x—  —  f  and  x  in  the  second  quadrant.  Find 
sin  2x.     (Sheffield.) 

10.  tan  z  =  ^  and  x  is  in  the  third  quadrant,  sec  y  =  —  y 
and  y  is  in  the  second  quadrant.  Find  cos  (x  —  2y);  cot  2x; 
sin  \y.     (Board.) 

11.  If  sin  z=-— — r,  find  tan  -.     (Harvard.) 

m2+n2  2 

12.  If  tan  ^  =  y,  find  the  values  of  sin  x  and  cos  x  in  terms  of  y. 
(Harvard.) 

13.  Prove  for  a  right  triangle  that  the  cosine  of  the  difference 
between  the  acute  angles  is  equal  to  twice  the  product  of  the 
two  legs,  divided  by  the  square  of  the  hypotenuse.     (Harvard.) 

2d  •    x 

14.  If  x  =  tan_1 -,  find  the  value  of  sin  -  in  terms  of  a 

\—az  & 

(consider  only  values  of  x  between  0°  and  90°,  and  values  of  a 
between  0  and  1).     (Harvard.) 

16.  Solve  the  equation 

tan-1  x+tan"1  2x  =  tan~1  3.     (Sheffield.) 

Let  a  =  tan-i  x,  /3=  tan-1  2x. 
Then     tan  (a+jS)  =3,  etc. 

16.  Prove  arc  tan  ^-+arc  tan  \  =  45? 

Trigonometric  Equations 

202.  Solve  the  following  equations  for  values  of  x 
between  0°  and  360°: 

1.  cos  2x-h3  sin  x  —  2 

Show  that  1-2  sin2  x +3  sin  x  =  2. 
Solve  for  sin  x.    Then  find  x. 

2.  cos  2z-f-cos  x  =  0 

3.  cos  x  cos  2z-f-2  cos3  x  =  0 
Solve  by  factoring. 


194  THIRD-YEAR  MATHEMATICS 

4.  tan  (^H-zj+tan  t-.—  x )  =4 

Expand  each  term. 

5.  sin  z+sin  2x=\ 

6.  cos  2x+ sin  a;  =  4  sin2  £ 
$7.  Solve  the  equation 

3  sec2  x— 7  tan2  z  =  tan  x. 

Obtain  all  solutions  for  x  between  0°  and  180°  and  give  the 
answers  to  the  nearest  degree.     (Yale.) 

$8.  Solve  the  equation 

sin  2z+J  =  sin  x-f-cos  x. 

Obtain  all  solutions  for  x  between  0°  and  180°  and  give  the 
answers  in  degrees.     (Yale.) 

$9.  Find  all  the  values  of  x  between  0°  and  360°  which  satisfy 
the  equation 

4  cos  2x+3  cos  x=l.     (Harvard.) 

$10.  Find  all  the  values  of  x  between  0°  and  360°  which 
satisfy  the  equation 

6  cos  2z+6  sin2  a:  =  5+sin  x, 
and  verify  your  results.     (Harvard.) 

Jll.  Find  all  the  values  of  x  between  0°  and  360°  which 
satisfy  the  equation 

3  cos  2rc+sin  x  (3  sin  x+5)  =  5.     (Harvard.) 

1 12.  Find  all  the  values  of  x  between  0°  and  360°  for  which 
2  sin  2x  =  cos  x.     (Harvard.) 

$13.  The  sum  of  the  tangents  of  the  acute  angles  of  a 
right  triangle  is  equal  to  4.  Find  the  values  of  these  angles. 
(Harvard.) 

$14.  Solve  the  equation 

cos  50+ cos  36=^2  cos  40.     (Princeton.) 


FUNCTIONS  OF  SEVERAL  ANGLES  195 

Summary 
203.  The  following  formulas  have  been  proved: 

1.  sin  (a+p)  =  sin  a  cos  p+cos  a  sin  p 

2.  cos  (a+P)  =cos  a  cos  p  — sin  a  sin  p 

3.  sin  (a  — P)  =sin  a  cos  p— cos  a  sin  p 

4.  cos  (a  — p)  =cos  a  cos  p+sin  a  sin  p 

5.  sin  a+sin  p=2  sin  \  (a+p)  cos|(a  — p) 

6.  sina-sin  p=2cos  *(a+p)  sin  £(a-p) 

7.  cosa+cos  p=2cos  *(a+P)  cos^(a-p) 

8.  cosa-cos  p  =  -2  sin^(a+p)  sin^(a-P) 
tan  a+tan  p 


9.  tan  (a+p) 
10.  tan  (a-p)  = 


1— tan  a  tan  p 
tan  a— tan  p 


1+tan  a  tan  p 

11.  sin  2a  =  2  sin  a  cos  a 

12.  cos  2a = cos2  a— sin2  a 

=2  cos2  a-1 
=  1-2  sin2  a 


2  tan  a  ._  a  fl+cos  a 

13.  tan  2a =z — i — =—  15.  cos 


l-tan2a  —         2 


\-4 


«.      .    a  ll— cosa       am  a  a 

14.  sm^  =  ^— 2—       16.  ten  g-*^ 


—cos  a 


1+cos  a 


204.  The  chapter  has  shown  how  to  solve  trigo- 
nometric equations.  Explain  your  method  of  solving 
such  equations. 

205.  Explain  how  to  prove  trigonometric  identities. 


CHAPTER  X 

BINOMIAL  THEOREM.    ARITHMETICAL  AND 
GEOMETRICAL  PROGRESSIONS 

Binomial  Theorem 

206.  The  binomial  theorem  enables  us  to  state  by  in- 
spection the  expansion  of  a  power  of  a  binomial.  By 
actually  multiplying,  the  following  identities  are  obtained: 

(a+&)2  =  a2+2a&+62 

(a+by  =  ai+3a'>b+3ab2+b3 
(a+&)4  =  a4+4a3&+6a262+4a&3+&4 

(a+b)5  =a5+5a4b+10a?b2+10a2b*+5ab*+V>y  etc. 

A  study  of  these  identities  brings  out  the  following  facts: 

1.  The  number  of  terms  in  the  right  member  is  one  greater 
than  the  exponent  of  the  binomial  in  the  left  member. 

2.  The  exponent  of  a  in  the  first  term  of  the  expansion 
is  the  same  as  the  exponent  of  the  binomial  and  decreases  by 
one  in  each  succeeding  term,  being  0  in  the  last  term. 

3.  The  exponent  of  b  increases  by  1  from  term  to  term, 
being  0  in  the  first  term  and  the  same  as  the  exponent  of  the 
binomial  in  the  last  term. 

Omitting  the  coefficients,  these  three  facts  give  the 
following  expansion  of  any  binomial,  as  {a+b),  raised  to 
any  positive  integral  power,  n: 

(a+b)n  =  an+(   )an~1b-\-(   )an~2b2+(   )a»-*&»+..., 

196 


BINOMIAL  THEOREM.     PROGRESSIONS  197 

The  coefficients  may  be  determined  by  the  following 
simple  device: 

Arrange  the  coefficients  of  (a+b)°,  (a-f-6)1,   (a+6)2, 

etc.,  as  in  the  form  given  in  Fig.  94.     Notice  that  each 

coefficient  in  this  arrangement  is  equal 

to  the  sum  of  the  coefficients  which  ±    1 

are  nearest  to  the  right  and  left  of  it  12    1 

13     3     1 
in  the  line  above.  14    6    41 

Fig.  94  is  known  as  Pascal's  tri-  1  5  10  10  5  1 
angle.*  Fig.  94 

Moreover,  after  the  second  term 
any  coefficient  may  also  be  determined  by  means  of  the 
coefficient  of  the  term  just  preceding,  according  to  the 
following  rule: 

Multiply  the  coefficient  of  the  preceding  term  by  the 
exponent  of  a  in  that  term  and  divide  the  product  by  the 
number  of  that  term. 

Thus  in  (a+6)5  the  coefficient  of  the  fifth  term  is 

— : —  =  5;  the  coefficient  of  the  third  term  is  -^-  =  10,  etc. 
4  2 

207.  The  binomial  theorem.  According  to  the  rules 
given  in  §  206  the  expansion  of  (a+b)n  takes  the  following 
form: 

(a+b)"=a"+na"-1b+n^~01)a»-*b* 

+n(n-lKn-2)an_3bS+ 

This  is  known  as  the  binomial  theorem  for  positive 
integral  powers.  The  theorem  is  assumed  without  proof. 
The  proof  is  usually  given  in  a  course  in  advanced 
algebra. 

*  See  First-Year  Mathematics,  pp.  200-201. 


198  THIRD-YEAR  MATHEMATICS 

208.  The  factorial  notation.  The  products  1  -2, 1  -.2.3, 
1*2* 3*4  ,....,  l*2a3*4  .  .  •  .  r,  are  called  factorial  2, 
factorial  3,  factorial  4,  ....  ,  factorial  r.  They  are 
usually  denoted  briefly  by  the  symbols:  2!,  3!,  4!, r! 

or  by  |2,  |3,  |4, ,  \r. 

Thus,  (a+6)w  =  a"+na»-16+^^a"--262 

.  n(n—  l)(n— 2)  „   „,  . 
+ 3^ -an~3V+ .... 

EXERCISES 

Expand  the  following  powers  to  four  terms: 

1.  (2x-Zyy 

In  this  example  a=2x,  b=(—3y),  n=7. 

3 
Hence  Vx-Zy)-*  =  (2xy+7(2x)*( -Sy)  +7-£(2x)H -32/)2+ 


^W(-3y)'+ 


=  2V-7-2«-3x6y4-21-25-32x6i/2-35-2433x4^H-.... 
=  128x7-1344^2/+6048x52/2-15120xV+.  • .  • 


'•  (;+?)' 


Herea=- ,  6=-|-,n=4 

•••  (2,4r=©4^©3(3f) 

4f ?)W)242©  »•+¥©'(? r 

_2*    4-23-3y2    6-22-3V     4-2-3Y     3V 
y*-1"      4y      +     42y2      +     4?y     +  44 


BINOMIAL  THEOREM.     PROGRESSIONS 


199 


■•  ®-"°) 


Let  a  =  T^>  b=  —bV  a 
'2a 


ba* 

a\4  . 


Then(|-6/a)'  =  (|)V4(|)V^) 


+V8®w>* 


1)5 
2> 


4.  (x-i/)8 

6.  (2a+4c)4 

6.  (3a2 -62)7 

7.  (3a264- 

8*   (4*-^ 

*•  K)* 

10.  (Vz+vV)5 

11.  (2+1/3)4 

12.   (^-^Y 

\  y      x  / 


24a4    4-23a3a*6     6-22aW 
6»  b6       +       64 

16a4  _32a3v/a    24a3 
68  65      +  62   


13.  Op-^Y 

\  Vbz      a  J 
14.  (z*+?/§)4 
15..  (a-3-6-3)4 
16.  (3a6"3-a-36)6 

»•  (S-f)' 
"•  (•'-*)' 

20.  (4af-a*6*)6 


209.  The  rth  term  of  (a+b)n.  The  number  of  the 
term  and  the  numerator  of  the  coefficient  of  the  same  term 
may  be  arranged  in  the  following  table : 


Number  of 
Term 

Numerator 

3 

4  ....... 

5 

6 

n(n  —  1) 
n(»— l)(n— 2) 

n(n-l)(n-2)(w-3) 
n(w-l)(n-2)(w-3)(n-4) 

200  THIRD-YEAR  MATHEMATICS 

From  a  study  of  this  table  show  that  the  numerator  of 
the  coefficient  of  the  rth  term  is 

n(w-l)(n-2)(n-3).  .  .  .  (n-r+2). 

Similarly,  find  that  the  denominator  of  the  coefficient 
of  the  rth  term  is  1-2-3-4-   ....  (r-1). 

Show  that  the  exponent  of  a  in  the  rth  term  is  n  —  r+1. 

Show  that  the  exponent  of  b  in  the  rth  term  is  r  —  1. 

Thus  the  rth  term  in  the  expansion  of  (a+6)n  is  given 
by  the  formula 

_  n(n-l)(n-2)  ....  (n-r+2)  _, 

lr         1l2._3-        (r-1)   a         °      * 


EXERCISES 

In  the  expansions  of  the  following  binomials  find  the  term 
called  for  in  each  case: 


•• in  (*r*>y 


find  the  fourth  term. 


Here  a  =  2x,  b=Fr,  n  =  6,  r  =  4 


20.23-x3 


20 


^(M) 


23x* 

\8 

find  the  fifth  term. 


3.  In  (^x+fty)7  find  the  sixth  term. 

4.  In  (%x—^y)10  find  the  fourth  term. 

5.  Write  the  last  three  terms  of  the  expansion  of  (4a3— a*s*)8. 


(Yale.) 


BINOMIAL  THEOREM.     PROGRESSIONS  201 

MISCELLANEOUS   EXERCISES 

J210.  The  following  problems  are  taken  from  college- 
entrance  examination  papers: 

1.  Expand  and  simplify  ( ^--77=  )  .     (Smith.) 

\y*    xv  —6/ 

2.  Write  the  last  three  terms  of  the  expansion  of  (4a*  —  a*x*)8. 
(Yale.) 

3.  Prove      that       (a+b)7-a7-b7  =  7ab(a+b)(a2+ab+b2)2. 
(Harvard.) 

/         1  \10 

4.  Find  the  fifth  term  of   (^4+fi)    and  reduce  to  the 

simplest  form.     (Dartmouth.)     ^  ' 


6.  In  the  expansion  of  ( 2x+—  J  the  ratio  of  the  fourth 
term  to  the  fifth  is  2:1.     Find  x.     (Princeton.) 

/    x        ^yV 

6.  Write  the  sixth  term  of  (  ^—  — )  .  (Pennsyl- 
vania.) 

7.  Find  and  simplify  the  twenty-third  term  in  the  expansion 

of  (¥-!)23-  (ComeU-) 

8.  If  the  middle  term  of  (  Sx— — r= )    is  equal  to  the  fourth 

/     ._        1    V  V         2Vx)  4- 

term  of  (  2 /  x+—^=  J  ,  find  the  value  of  x.     (M.I.T.) 

9.  Write  the  first  term  of  (x*  —  x~*)s  which  in  its  simplest 
form  has  a  negative  exponent.     (Board.) 

10.  Find  the  coefficient  of  x4  in  (  2x2— —  J  . 

11.  Show  ttfat  the  coefficient  of  the  middle  term  of  (l-fz)16 
is  equal  to  the  sum  of  the  coefficients  of  the  eighth  and  ninth 
terms  of  (1-fx)15.     (Princeton.) 


202  THIRD-YEAR  MATHEMATICS 

12.  In  the  expansion  of  (2x—Sx~1)8  find  that  term  which 
does  not  contain  x.     (Princeton.) 

13.  Write  the  sixth  term  in  the  expansion  of 

10 


W 


64aW  ,  .  /     o  (Yale0 


81m2n8     \     m" 
14.  Find  the  third  and  fifth  terms  in  the  expansion  of 

(1- ■/£)•.     (Sheffield.) 

(2a;2     3  \  8 
— — —J    find  the  coefficient  of  x4. 

16.  In  the  expansion  of  ( a+- )  write  the  term  which 
does  not  contain  a. 

17.  Find  the  middle  term  in  the  expansion  of  ( -+- )    . 

\b    a) 

Arithmetical  Progression 

211.  Arithmetical  progression.  A  succession  of  num- 
bers formed  according  to  a  definite  law  is  called  a  series. 
Thus  the  expression  2+4+6+8....  is  a  series,  the 
law  of  formation  being  that  any  term  in  the  series  is 
obtained  by  adding  the  fixed  number,  2,  to  the  preceding 
term. 

An  arithmetical  progression  is  a  succession  of  numbers 
in  which  each  term  after  the  first  may  be  found  by  adding 
a  constant  number  to  the  preceding  term. 

EXERCISES 

1.  Show  that  the  following  are  examples  of  arithmetical 

progressions:  3,  5,  7,  9 ;  84,  74,  64 ;  the  logarithms,  to 

the  base  3,  of  the  numbers  3,  9,  27,  81 

2.  Show  that  the  equation  a—b  =  b—c  expresses  the  fact 
that  three  numbers  a,  b,  and  c  are  in  arithmetical  progression. 

212.  Elements.  In  general,  the  form  of 'an  arith- 
metical progression  is 

a,  (a+d),  (a+2d),  (a+3d),.... 


BINOMIAL  THEOREM.     PROGRESSIONS 


203 


The  first  term  is  denoted  by  a, 
the  constant  difference  by  d, 
the  number  of  terms  considered  by  w, 
the  nth  term  by  I, 
and  the  sum  of  n  terms  by  s. 

The  numbers  a,  d,  n,  I,  and  s  are  the  elements  of  the 
arithmetical  progression. 

213.  Relations  between  the  elements.  The  table, 
Fig.  95,  shows  one  of  the  relations  between  the  elements 
of  an  arithmetical  progression. 


Number  of  Term 

Term 

Second 

a+d 

Third 

a+2d 

Fourth 

a+Sd 

Fifth 

a+4d 

Tenth 

a+9d 

Fifteenth 

a+Ud 

nth 

a  +  (n-l)d 

.-.    l  =  a+(n-l)d 
Fig.  95 


Another  relation  may  be  obtained  as  follows : 

s  =  a+(a+d)  +  (a+2d)+  ....  a+(n-l)d 
Similarly, 

■  8  =  l  +  (l-d)  +  (l-2d)+  ....  l-(n-l)d 

Adding, 

2s=(a+Z)  +  (a+Z)  +  (a+Z)+  ....  +(a+l) 
Combining  terms, 

2s  =  n(a+l) 


s=^(a+l) 


The  two  relations  just  established  enable  us  to  find  the 
values  of  two  of  the  elements  if  the  other  three  are  known. 


204  THIRD-YEAR  MATHEMATICS 

EXERCISES 

Solve  the  following  problems: 

1.  Find  the  tenth  term  of  the  series  7+10+13 

Let  a  =  7,  d  =  S,  n  =  10. 

Substitute  these  values  in  the  equation  l  =  a  +  (n  —  l)d,  and  find 
the  required  value  of  I. 

2.  Find  the  seventh  term  and  the  sum  of  seven  terms  of  the 
series  2+4+6 

3.  If  a=7,  d=  —2,  and  w  =  8,  find  s  and  I. 

4.  If  Z=30,  n  =  9,  and  s  =  162,  find  a  and  d. 

5.  The  fourth  term  of  an  arithmetical  progression  is  11  and 
the  fourteenth  term  is  —39.    Find  the  common  difference. 

6.  The  sum  of  the  second  and  twentieth  terms  of  an  A.P.  is 
10,  and  their  product  is  23|~|.  What  is  the  sum  of  16  terms  ? 
(Pennsylvania.) 

7.  Given  Z  =  23,  d=2,  s  =  143.     Find  a  and  n. 
By  substituting  for  I,  d,  and  s  the  given  values, 

23  =  <z+(n-l)2 

143=|(a+23) 


or  a+2n  =  25 

an +23n  =  286 

Solving  the  first  equation  for  a  and  substituting  into  the 
second, 

(25-2n)n+23n=286 
or  2n2-48n +286  =  0 

.'.  n  =  ll,  33 

Evidently  there  are  two  progressions :  3,  5,  7,  9, and 

-41,  -39,  -37.... 

8.  Given  o  =  5,  d  =  3,  and  s=185.  Find  I  and  n. 

9.  Given  Z=33,  s=  152,  and  d=4.  Find  a  and  n. 


BINOMIAL  THEOREM.     PROGRESSIONS  205 

10.  Find  the  sum  of  all  even  integers  from  1  to  100. 

11.  Find  the  sum  of  all  integers  divisible  by  3,  between  0 
and  320. 

12.  Find  the  sum  of  all  positive  integers  of  three  digits  which 
are  multiples  of  9. 

13.  Find  the  nth  term  and  the  sum  of  n  terms  in  the  progres- 
sion 1,  3,  5,  7 

14.  Find  the  first  term  if  the  common  difference  is  5  and 
the  twenty-seventh  term  is  139. 

15.  How  many  numbers  divisible  by  6  are  between  0  and 
200?  , 

16.  A  man  agreed  to  dig  a  well  at  the  following  rate:  for 
the  first  yard  he  was  to  receive  $8  and  for  every  succeeding 
yard  $2  more.  If  the  well  was  27  yd.  deep,  how  much  did  he 
receive  ? 

17.  A  man  buys  a  house  and  lot  for  the  sum  of  $6,200.  He 
agrees  to  pay  $600  at  the  end  of  the  first  year,  $650  at  the  end 
of  the  second  year,  etc.  How  many  years  will  it  take  him  to  pay 
for  the  property  ? 

18.  The  sum  of  the  first  eight  terms  of  an  arithmetical 
progression  is  64  and  the  sum  of  the  first  18  terms  is  324.  Find 
the  series.     (Princeton.) 

19.  A  man  takes  a  position  with  the  understanding  that  he 
will  receive  $800  the  first  year  with  an  increase  of  $50  each  suc- 
ceeding year  for  the  next  20  years.  What  is  his  salary  during 
the  fifteenth  year  ? 

20.  In  a  potato  race  45  potatoes  are  placed  in  a  straight 
line  with  a  basket  and  3  ft.,  6  ft.,  9  ft.,  12  ft.,  etc.,  from  it.  What 
is  the  total  distance  a  boy  must  run  to  carry  the  potatoes  to  the 
basket  one  at  a  time  ? 

21.  A  ball  rolling  down  an  inclined  plane  goes  8  ft.  the  first 
second.    In  each  second  thereafter  it  passes  over  16  ft.  more 


206  THIRD-YEAR  MATHEMATICS 

than   in  the  preceding  second.    How  far  will  it  roll  in   12 
seconds  ? 

22.  A  body  starting  from  rest  is  observed  to  fall  16.08  ft. 
in  the  first  second,  48.24ft.  in  the  second,  80.40  in  the  third, 
etc.  How  far  does  it  fall  in  12  seconds?  in  15  seconds?  in 
t  seconds  ? 

23.  The  sum  of  three  terms  of  an  arithmetical  progression 
is  33.  The  square  of  the  last  term  exceeds  the  sum  of  the  squares 
of  the  first  two  by  11.    What  are  the  numbers  ? 

24.  A  bullet  is  fired  directly  upward  with  a  speed  of  1,800  ft. 
a  second.  As  it  goes  up  its  speed  decreases  and  as  it  comes  down 
its  speed  increases  by  32  ft.  a  second.  How  high  will  it  rise? 
In  what  time  will  it  reach  the  ground  ? 

214.  Arithmetical  means.  The  terms  of  an  arith- 
metical progression  between  any  two  other  terms  are 
called  arithmetical  means. 

For  example, 

5  is  an  arithmetical  mean  between  2  and  8; 

9,  5,  1,  and  —3  are  four  arithmetical  means  between 
13  and  -7. 

Between  two  given  numbers  one  or  more  arithmetical 
means  may  be  inserted. 

Thus,  if  three  arithmetical  means  are  to  be  inserted 
between  5  and  69,  we  have  the  series  5+  .  .  .  +69,  the 
three  dots  indicating  the  three  means  to  be  inserted. 

Hence  a  =  5,  Z  =  69,  n  =  5,  and  d  is  to  be  found. 

From  the  formula  l  =  a+(n—  l)d,  we  have 

69  =  5+4d, 

.'.     d=16. 
i 

Hence  the  three  arithmetical  means  between  5  and  69 
are  21,  37,  53. 


BINOMIAL  THEOREM.     PROGRESSIONS  207 

EXERCISES 

1.  Insert  8  arithmetical  means  between  —5  and  —3. 

2.  Insert  4  arithmetical  means  between  9  and  11. 

3.  Find  the  arithmetical  mean  between  15  and  7. 

4.  How  many  arithmetical  means  must  be  inserted  between 
249  and  15  to  make  the  sum  of  the  series  equal  to  1,995  ? 

5.  Insert  6  arithmetical  means  between  8  and  3. 

6.  Arithmetical  means  are  inserted  between  1  and  21  so  that 
the  sum  of  these  means  is  132.  Find  the  first  two  of  them. 
(Board.) 

Geometrical  Progression 

215.  Geometrical  progression.  A  succession  of 
numbers  in  which  any  term  after  the  first  is  obtained  by 
multiplying  the  preceding  term  by  a  fixed  number  is  a 
geometrical  progression. 

The  fixed  number  is  called  the  common  ratio. 
The  general  form  of  geometric  progression  is 
a,  ar,  ar2,  ar3. . . ., 
where  a  is  the  first  term  and  r  the  common  ratio. 

EXERCISES 

1.  Show  that  the  following  are  examples  of  geometrical  pro- 
gressions and  find  the  common  ratio : 

1.  3,6,12,24.... 

2.  27,  -9, +3,  -1... 

2.  Show  that  three  numbers,  a,  b,  and  c,  are  in  geometrical 

progression  if  the  ratio  ?  is  equal  to  the  ratio  -,  i.e.,  if  7  =  -. 
o  c  o    c 

216.  Elements   of   a  geometrical  progression.    The 

first  term,  a,  the  nth  term,  I,  the  number  of  terms,  n, 
the  common  ratio,  r,  and  the  sum  of  n  terms,  s,  are  the 
elements  of  the  geometrical  progression. 


208 


THIRD-YEAR  MATHEMATICS 


217.  Relations  between  the  elements.  The  table, 
Fig.  96,  shows  how  one  of  the  relations  between  the  ele- 
ments may  be  found. 


Number  of  the 
Term 

Term 

Second 

Third 

ar 

ar2 

ar3 

ar4 

ar11 

ar«-i 

Fourth 

Fifth 

Twelfth 

nth 

.-.    1  = 

Fig 

ar"'1 

96 

A  formula  for  the  sum  of  n  terms  is  worked  out  as 
follows : 

Let    s  =  a  -\-ar  +ar2-\-ar*+  .  .  .  .  arn~2-\-arn~l. 
Multiplying  by  r, 

rs  =  ar-\-ar2+arz+arA+  ....  arn~l-\-arn. 

Subtracting  the  second  equation  from  the  first, 
s  —  rs  =  a  —  arn, 
or  (1—  r)s  =  a(l  —  rn), 

a(l-r") 


s  = 


1-r     * 


EXERCISES 


1.  Find  the  twelfth  term  of  the  progression  1, 
What  is  the  sum  of  the  first  8  terms  ?    - 

2.  Given  a=  16,  r=  —  -,  1=  —-.    Find  n  and  s. 

Z  o 

3.  Find  the  sum  of  10  terms  of  the  progression 
27,  -9,  3,  -1.... 

4.  Given  a  =  5,  n=  10,  s  =  50.     Find  I  and  r. 


1    1 
2'4*   *' 


BINOMIAL  THEOREM.     PROGRESSIONS  209 

6.  Find  the  eighth  term  of  the  series  -jr-  — 5- +4— 3+ 

6.  The  fourth  term  of  a  geometrical  progression  is  32,  the 
eighth  term  is  512.     Find  the  tenth  term. 

7.  The  sum  of  the  first  and  third  terms  of  a  geometrical  pro- 
gression is  40  and  the  second  term  is  16.     Find  each  term. 

8.  Three  numbers  are  in  geometrical  progression.  The 
second  is  32  greater  than  the  first  and  the  third  96  greater  than 
the  second.     Find  the  numbers. 

9.  A  chain  letter  is  sent  by  a  person  to  two  friends  with  the 
request  that  each  send  a  copy  of  the  letter  to  two  friends,  with 
a  request  that  they  in  turn  send  a  copy  to  each  of  two  friends, 
etc.  After  12  sets  of  letters  have  been  sent,  how  many  copies 
of  the  original  letter  have  been  made  ? 

10.  An  elastic  ball  bounces  to  three-fourths  of  the  height 
from  which  it  falls.  If  it  is  thrown  up  from  the  ground  to  a 
height  of  15  ft.,  find  the  total  distance  traveled  before  it  comes 
to  rest.     (Board.) 

11.  The  sum  of  the  first  8  terms  of  a  geometrical  progression 
is  seventeen  times  the  sum  of  the  first  4  terms.  Find  the  value 
of  the  common  ratio.     (Board.) 

12.  The  sum  of  the  first  10  terms  of  a  geometrical  progres- 
sion is  equal  to  244  times  the  sum  of  the  first  5  terms,  and  the 
sum  of  the  fourth  and  sixth  terms  is  135;  find  the  first  term  and 
the  common  ratio.     (Princeton.) 

13.  A  capital,  c,  is  placed  on  interest  at  r  per  cent,  com- 
pounded annually.  What  is  the  amount  at  the  end  of  the  first 
year  ?    Second  year  ?  etc. 

Show  that  at  the  end  of  the  first  year  the  amount  will  be 

c+m=c(1+wo)- 

Show  that  at  the  end  of  the  second  year  the  amount  will  be 

<1+m)H1+m)m~i1+m)(1+m)=i1+m)2- 

Show  that  at  the  end  of  the  third  year  the  amount  will  be 


< 


r  V 

+ioo)  - etc- 


210  THIRD-YEAR  MATHEMATICS 

14.  What  is  the  amount  in  exercise  13  at  the  end  of  the  nth 
year? 

15.  What  is  the  amount  of  $1Q0  at  6  per  cent,  compounded 
annually,  at  the  end  of  2  years  ?     10  years  ?    n  years  ? 

16.  What  is  the  amount  of  $25  at  3f  per  cent,  compounded 
annually,  at  the  end  of  6  years  ?     15  years  ? 

$17.  A  rubber  ball  falls  from  a  height  of  40  in.  and  on  each 
rebound  rises  40  per  cent  of  the  previous  height.  Find  by 
formula  how  far  it  falls  on  its  eighth  descent.        t 

$18.  An  elastic  ball  drops  from  a  height  of  16  ft.  on  a 
hard  pavement  and  rebounds  again  and  again.  If  the  time 
of  each  rise  is  f  of  the  time  of  the  preceding  fall,  show  that 
the  time  during  which  bouncing  continues  cannot  exceed 
7  seconds  (the  time  required  for  the  first  fall  is  1  second). 
(Harvard.) 

$19.  A  rubber  ball  is  dropped  from  a  height  of  20 
feet.  After  each  rebound  it  rises  to  T\  of  the  height  from 
which  it  fell.  How  far  does  it  travel  before  it  comes  to  rest  ? 
(Harvard.) 

218.  Geometrical  means.  The  terms  of  a  geometrical 
progression  included  between  two  other  terms  are  called 
geometrical  means. 

In  the  progression  1,  3,  9,  27  the  terms  3  and  9  are 
two  geometrical  means  between  1  and  27. 

The  following  example  illustrates  the  method  of  insert- 
ing a  number  of  geometrical  means  between  two  given 
numbers : 

Insert  3  geometrical  means  between  2  and  32. 
Here  a  =  2,  n  =  5,  Z  =  32 

.\32  =  2r4 
.\r  =  2 
The  geometrical  means  are  4,  8,  and  16. 


BINOMIAL  THEOREM.     PROGRESSIONS  211 

EXERCISES 

1.  Insert  three  geometrical  means  between  486  and  6. 

2.  Find  the  geometrical  mean  between 
the  two  segments,  Fig.  97,  into  which  the 
altitude  from  the  vertex  of  the  right  angle 
divides  the  hypotenuse. 

3.  Find  the  geometrical  mean  between  F      97 
8  and  32. 

4.  Insert  three  geometrical  means  between  §  and  12. 

a     c 

5.  If  y  =-,  prove  that  ab+cd  is  a  mean  proportional  between 

a2-\-c2  and  b2+d2.     (Princeton.) 

MISCELLANEOUS   EXERCISES 

J219.  Solve  the  following  problems: 

1.  Find  the  sum  of  n  terms  of  the  series 

<>-'>+(J-£)+(g-S)+-  tfw 

2.  The  sum  of  three  numbers  in  geometrical  progression  is 
70.  If  the  first  be  multiplied  by  4,  the  second  by  5,  and  the  third 
by  4,  the  resulting  numbers  will  be  in  arithmetical  progression. 
Find  the  three  numbers.     (Board.) 

3.  If  r — ,  -7  ,  and  r —  are  in  arithmetical  progression, 

o    a     £o  o — c 

show  that  a,  b,  and  c  are  in  geometrical  progression.     (Yale.) 

4.  An  arithmetical  progression  and  a  geometrical  progres- 
sion have  the  same  first  term,  3,  equal  third  terms,  and  the 
difference  of  the  second  terms  is  6.     Determine  the  progressions. 

5.  The  sum  of  9  terms  of  an  arithmetical  progression  is  46; 
the  sum  of  the  first  5  terms  is  25.  Find  the  common  difference. 
(Vassar.) 

6.  The  difference  between  two  numbers  is  48.  Their  arith- 
metical mean  exceeds  the  geometrical  mean  by  18.  Find  the 
numbers.    (Smith.) 


212  THIRD-YEAR  MATHEMATICS 

Infinite  Geometrical  Series 

220.  Infinite  geometrical  series.  We  have  learned 
how  to  find  the  sum  of  n  terms  of  a  geometrical  series, 
n  being  a  finite  number.     If  the  number  of  terms  in  the 

series  a+ar-\-ar2+ is  unlimited,  it  is  called  an  infinite 

geometrical  series.  If  the  sequence  of  the  partial  sums, 
a,  a-\-ar,  a-\-ar-\-ar2,  a-{-ar-\-ar2-{-ar3,  etc.,  approaches  a 
definite  finite  number,  S,  as  a  limit,  we  say  S  is  the  sum 
of  the  infinite  series 

a+ar +ar2-|-ar3+ 

For  example,  to  find  the  sum  of    A '  b      »  Epc 

the    series    l+|+J+|+....     we       t"ut±K-~-i 

may    represent    the    partial   sums       i—  — 1-&*  \-lA\% 1 

graphically,  Fig.  98.  Fig.  98 

Let  AB  =  1  =BC.     Then  AD  =  1+|. 

Bisecting  the  remainder,  DC, 

Bisecting  the  second  remainder,  EC, 

AF=i+Hi+i 

As  the  process  of  increasing  the  number  of  terms  con- 
tinues, the  partial  sum  increases,  approaching  the  num- 
ber 2  in  such  a  way  that  it  can  be  made  to  differ  from  2  by 
less  than  any  assigned  number,  however  small. 

Hence  2  is  said  to  be  the  sum  of  the  infinite  series 

The  series  is  a  convergent  series. 
Another  example  of  a  convergent  infinite  geometrical 
series  is  found  in  a  recurring  decimal  fraction. 

The  number  .  333 ....  is  really  a  brief  way  of  writing 

TU+TTnr+TUinT+TiFiFinr » 

a  geometrical  series  in  which  a  =  t3q-  and  r  =  T1Tr. 


BINOMIAL  THEOREM.     PROGRESSIONS  213 

Show  that  the  sum  of  this  series  is  J. 

The  two  preceding  examples  are  particular  cases  of  an 
infinite  geometrical  series  whose  ratio  r  is  numerically 
less  than  1. 

If  in  the  series  a-\-ar+ar2+ars-\- . . 
we  take  r  numerically  equal  to  1,  we  have  either 

a+a+a+a+ , 

or  a—a+a—a-J- . . . . 

In  the  first  case  the  sum  increases  without  bound  as 
the  number  of  terms  increases  indefinitely.  In  the  second 
case  the  partial  sums  a,  a— a,  a— a + a,  etc.,  have  alter- 
nately the  value  a  or  0.  Hence  in  either  case  the  series 
has  no  definite  sum. 

The  question  as  to  the  sum  of  an  infinite  geometrical 
series  may  be  considered  by  making  a  study  of  the 
formula  for  the  sum  of  n  terms, 

1  —  rn     a  —  arn       a        arn 
sn  =  a 


1  — r       1  —  r      1  —  r    1  —  r 

a         arn 


1.  If  r>l,   then  rn  increases  without  bound  as  n 
increases  indefinitely.     This  is  expressed  symbolically  by 

the  statement  nm  v  )  =  oo  which  is  read, 

"The  limit  of  rn,  as  n  increases  indefinitely,  is  infinite." 
Hence  sn  also  increases  numerically  without  bound 
and  the  series  has  no  sum. 

2.  If  r<l,  lim  (rn)=0, 


214 


THIRD-YEAR  MATHEMATICS 


For  example,  if  r  =  J,  the  values  of  rn  for  w=l,  2, 

3,  4, are  respectively  the  numbers  of  the  sequence 

h  h  h  tV >  *£  r=  A,  we  have  the  sequence  .  1,  .01, 

.001,  .0001,.... 


lim  /   a        arn\=    a     , 
n^»co\l  —  r    1  —  r)     1  —  r' 


Hence  the  infinite  geometrical  series 
a+ar-\-ar2-\- . . . .  has  a  sum,  s,  given  by  the  formula, 

s=lim(Sn)=   «    ,  if  r<i. 


n->00 


1-r 


EXERCISES 

Find  the  sum  of  each  of  the  following  infinite  series: 


1.3+1+1+1+ 


Let  a  =  3,  r 
Thens 


1 

3" 
a  3 


1-r  1     2' 

3 


2.  10+5+2J+1J.. 

3.  _3+1-l  +  l... 


4.   -2-1-!. 
4    32 


5.4+2+1+.. 
3^3^3^ 


6.6-4+3.... 
(Princeton.) 


9.  i-M-1.. 

2^4    8 


8.  1.35+0.045+0.0015+.. 

(Harvard.) 

Also  find  the  sum  of  the  positive  terms. 

10.  In  a  geometrical  progression  the  sum  to  infinity  is  64 
times  the  sum  to  6  terms.  What  is  the  common  ratio  ?  (Prince- 
ton.) 

11.  The  first  term  of  a  geometrical  progression  is  225  and 
the  fourth  term  is  14J-.    Find  the  series  and  the  sum  to  infinity. 


BINOMIAL  THEOREM.     PROGRESSIONS  215 

Find  the  limiting  value  of  each  of  the  following  repeating 
decimals: 

12.  .1666....  14.  1.2121.... 
Leta  =  ^,  r«i.  15.  .83333.... 
Find  s  and  add  .  1  to  the  result.  16.  .  234234 ■ 

13.  .363636....  17.  .23737 

221.  Historical  note.  The  arithmetical  and  geometrical 
progressions  are  among  the  oldest  topics  of  all  mathematics. 
Problems  leading  to  both  kinds  of  progression  are  found  in  the 
oldest  extant  historical  document,  the  papyrus  of  Ahmes. 
The  forms  of  progression  called  for  by  the  problems  mentioned 
in  this  manuscript  are  very  far  from  the  simplest,  indicating 
that,  for  thousands  of  years  before  the  Christian  era,  Egyptian 
scholars  had  studied  these  progressions  and  by  2000  to  1700  B.C. 
they  had  attained  to  an  advanced  stage  of  knowledge  of 
them. 

The  Babylonians  made  use  of  both  forms  of  progression  in 
recording  the  phases  of  the  moon,  and  the  Greeks  were  zealous 
students  of  the  progressions.  The  theory  was  very  greatly 
advanced  by  the  Pythagoreans  in  connection  with  their  work  in 
figurate  numbers,  which  was  also  a  favorite  subject  of  the  school 
of  Plato.  Archimedes  was  even  well  acquainted  with  the  laws 
of  summation  of  the  progressions.  Heron  made  extended  prac- 
tical use  of  the  laws.  Hypsicles  and  Nicomachus  both  studied 
and  taught  the  topics  very  fully. 

The  Hindus  never  advanced  beyond  the  attainment  of  the 
Greeks.  They  solved  a  few  problems  requiring  a  knowledge 
of  the  progressions,  and  regarded  the  study  as  belonging  to 
arithmetic.  The  Arabs  advanced  considerably  beyond  the 
Hindus.  They  were  in  possession  of  the  completed  theory  of 
these  topics,  if  we  may  judge  from  the  work  of  Leonardo  of 
Pisa,  who,  in  his  liber  abaci  of  1202  a.d.,  brought  together  what 
was  known  by  the  Greeks  and  Arabs  and  made  it  available  for 
European  scholars.  Leonardo  even  made  summing  of  these 
series  one  of  the  nine  fundamental  processes  of  arithmetic,  thus 


216  THIRD-YEAR  MATHEMATICS 

putting  what  he  called  progressio  on  a  par  with  additio,  subtract™, 
multiplication  etc. 

The  progressions  formed  a  basis  for  the  ancient  Greek  method 
<of  exhaustions,  for  Cavalieri's  theory  of  indivisibles,  see  §  302, 
for  the  later  summation  schemes  of  the  infinitesimal  calculus, 
and  out  of  the  association  of  an  arithmetical  and  a  geometrical 
series  term  by  term  grew  the  subject  of  logarithms.  The  topics 
are  therefore  important  mathematically  both  for  the  problems 
they  aid  in  solving  and  for  the  large  amount  of  mathematical 
theory  and  methodology  they  have  aided  in  developing.  See 
Tropfke,  Band  II,  S.  309  ff. 

Summary 

222.  The  chapter  has  taught  the  meaning  of  the  fol- 
lowing terms: 

binomial  theorem  geometrical  means 

arithmetical  progression  infinite  progression 

arithmetical  means  elements  of  a  progression 

geometrical  progression  convergent  series 

223.  The  following  are  the  typical  problems  solved 
in  the  chapter: 

1.  To  expand  a  power  of  a  binomial  to  a  given  number 
of  terms. 

2.  To  find  a  required  term  in  the  expansion  of  a  power 
of  a  binomial. 

3.  Given  ;three  elements  of  a  progression,  to  find  the 
remaining  two. 

4.  To  insert  a  number  of  arithmetical,  or  geometrical, 
means  between  two  given  numbers. 

5.  To  find  the  sum  of  n  terms  of  a  progression. 

6.  To  find  the  sum  of  an  infinite  geometrical  progres- 
sion. 


BINOMIAL  THEOREM.     PROGRESSIONS  217 


224.  The  following  formulas  have  been  developed. 
1.  (fl+&)n  =  gn+n(g)n~16+n(1yi<~21)gn~V+.. . . 

9    /  _w(n-l)(n-2)  ....  (n-r+2)       h-i      i 
2'  *r"  1    ..  2    ,  3    .  .  .  .    (r-1)      °         b 

3.  l  =  a+(n-l)d  5.  Z  =  arn_1 


n 


4.  »==(a+/)  6.  s  = 


(l-rn) 


7.  5=^ ,  if  r<l,  and  if  the  number  of  terms  is 

1  —  r 


unlimited. 


CHAPTER  XI 


SYSTEMS  OF  EQUATIONS  IN  TWO   UNKNOWNS 
INVOLVING  QUADRATICS 

Graphs  of  Quadratic  Equations  in  Two  Unknowns 

225.  General  quadratic  equation.    The  most  general 
quadratic  equation  in  two  unknowns  is 

ax2+bxy+cy2+dx+ey+f=0. 

If  o  =  c  =  0,  the  equation  takes  the  form 
ax2+dx+ey+f=0. 
Solving  for  y, 


y 


— -t2  — 


-x— 


f 


This  expression  is  of  the  form  of  the  quadratic  function 

ax2+bx+c, 

the  graph  of  which  is  known  to  be  a  parabola,  §  13. 

226.  Circle.  It  will  be  seen  that  the  parabola  is  not 
the  only  curve  representing  an  equation  of  the  second 
degree  in  two  unknowns.  The  following  examples  serve 
as  illustrations: 

1.  Let  0,  Fig.  99,  be  the  center 
of  a  circle  with  radius  r. 

For  any  point  on  the  circle,  as 
P,  the  distance  PO  is  constant  and 
equal  to  r. 

Let  x  and  y  be  the  co-ordinates 
of  P. 

By  the  theorem  of  Pythagoras, 

*2+y2=r2 

218 


Fig.  99 


EQUATIONS  IN  TWO  UNKNOWNS 


219 


This  is  the  equation  of  the  circle  of  center  0  and  radius  r. 
The  graph  of  the  equation,  being  a  circle,  is  easily  drawn  with 
the  compasses. 

2.  Let  0,  Fig.  100,  be  the  center  of  a  circle  with  radius  r. 

Let  the  co-ordinates  of  0  be  h  and  k,  and  of  P  be  x  and  y. 


R 


Fig.  100 
Then  QP2+W  =  r2. 

Since  QP=y—k,  and  OQ  =  x—h,  this  equation  may  be 
written 

(x-h¥+(y-k¥=rK 

This  is  the  equation  of  a  circle  whose  radius  is  r  and  whose 
center  has  the  co-ordinates  h  and  k.  Expanding  the  squares 
of  the  binomials  the  equation  of  the  circle  takes  the  form 


x2+y2-2hx-2ky+(h2+k2-r2)  =0. 


(1) 


This  is  an  equation  of  the  second  degree,  but  not  of  the  form 
of  the  general  equation,  §  225,  for  it  contains  no  xy-term,  and 
the  coefficients  of  x2  and  y2  are  equal.  If,  in  the  general  equa- 
tion, we  put  b  =  0,  a=C7±0,  show  that  it  may  be  reduced  to  an 
equation  of  the  form  of  equation  (1), 

EXERCISES 

1.  Show  that  the  following  are  equations  of  circles  and  deter- 
mine the  radius  of  each: 

x2+2/2  =  9.  z2+2/2  =  5;  2x2+2i/2=8. 


220 


THIRD-YEAR  MATHEMATICS 


2.  Give  the  equations  of  circles  having  the  center  at  the 
origin  and  a  radius  equal  to  16;  a;   V2. 

3.  Compare  the  equations  in  exercise  1  with  the  general 
quadratic  iiv  §  225  and  in  each  case  determine  the  values  of  the 
coefficients. 

227.  Graph  of  the  equation 

x2  +y2  -  2hx  -  2ky + (h2+k2  -  r2)  =  0. 

If  particular  values  are  assigned  to  h  and  k  and  r, 
as  /i  =  4,  fc  =  3,  r=Vl3,  this  equation  reduces  to 

x2+y2-8x-6y+12=0, 
or  y2-$y+(x2-8x+12)=0. 

1.  Solving  for  y, 


6*l/-4s»+323!-12_3d,T/_a,+te_8 


2.  The  table,  Fig.  101,  gives  the  values  of  y  corre- 
sponding to  assumed  values  of  x. 


X 

y 

0 

1 

2 

3 

4 
5 
6 

7 
8 

3=*=v  —  3,  imaginary 
3±2,  or  5,  1 
3±3,  or  6,  0 
3±l/l2,  or  6.5,  -.5 
3^/13,  or  6.6,  -.6 
3=^12,  or  6.5,  -.5 
3±3,  or  6,  0 
3±2,  or  5,  1 
3±V  —  3,  imaginary 

Fig.  101 

3.  By  plotting  the  points  determined  by  the  values  in 
Fig.  101,  we  obtain  the  required  graph. 


GIRO LA  MO       CARDANO 

GIROLAMO  CARDANO  (English,  Cardan)  was  born 
at  Pavia  in  1501  and  died  at  Rome  in  1576.  That 
Cardan  was  one  of  the  leading  algebraists  of  all  time 
there  is  no  question.  The  chief  mathematical  work  of  Cardan 
is  the  Ars  magna,  published  in  1545.  It  was  both  epoch- 
making  and  a  masterpiece  of  algebraic  scholarship.  It  gave 
to  the  world  for  the  first  time  the  solution  of  both  the  cubic 
and  the  biquadratic  equations. 

That  Cardan  discovered  neither  of  these  solutions  there  is 
no  doubt.  Cardan  never  claimed  them  as  his  discoveries. 
The  credit  of  discovery  of  the  solution  of  the  cubic  that  is 
commonly  referred  to  in  American  texts  as  Cardan's  solution 
is  quite  generally  regarded  as  due  to  Tartaglia.  The  facts 
seem  to  be  that  one  Scipio  Ferro  in  some  unknown  way  came 
into  possession  of  the  solution  of  the  form  x3+ax  =b,  and  told 
it  to  numerous  friends,  among  them  one  Fiori  (Floridas). 
According  to  Cardan  this  occurred  in  1515,  and  according  to 
Tartaglia  as  early  as  1506.  In  1535  Fiori  used  this  knowledge 
to  propose,  in  accord  with  a  custom  of  the  day,  thirty  chal- 
lenge problems  to  Tartaglia,  then  a  mathematical  teacher  of 
repute.  Tartaglia  knew  well  that  all  the  problems  would  lead 
to  the  form  x?+ax  =b,  and  he  bestirred  himself  for  a  long  time 
vainly  to  discover  the  solution;  but  eight  days  before  the 
contest  he  says  he  succeeded.  The  next  day  he  discovered 
the  solution  formula  for  x*=ax+b.  In  the  contest  he  solved 
all  of  Fiori's  problems  in  some  two  hours. 

Cardan,  then  a  professor  of  mathematics  at  Bologna, 
besought  Tartaglia  to  publish  his  solution.  Tartaglia  refused. 
Later  in  obscure  verse  he  did  hint  at  it.  Cardan  agreed  not 
to  divulge  Tartaglia's  solution  if  Tartaglia  would  impart  it  to 
him.  A  letter  of  Tartaglia's  later  explained,  again  obscurely, 
some  things  hinted  at  in  the  verse.  Then  in  1545  when 
Cardan's  Ars  magna  appeared  it  contained  the  correct  solu- 
tion with  proof,  of  the  cubic  and  the  biquadratic  equations, 
with  due  credit  given  both  to  Tartaglia  and  to  Ferari,  who  was 
discoverer  of  the  solution  of  the  biquadratic.  Tartaglia  was 
now  greatly  annoyed  amd  accused  Cardan  of  violating  a 
solemn  pledge  of  secrecy.  Later  Cardan  showed  that  in  the 
literary  remains  of  Professor  Ferro  was  the  solution  of  the 
cubic  precisely  identical  with  that  of  Tartaglia  and  dating 
thirty  years  earlier  than  Tartaglia's  contest. 

Debate:  Is  Cardan  justly  chargeable  with  literary  theft 
for  publishing  Tartaglia's  solution,  which  was  identical  with 
Ferro's  of  thirty  years'  earlier  date  ? 

[See  Tropfke,  Geschichte  der  Elementar-Mathematik,  Band  I, 
S.  274  ff.] 


EQUATIONS  IN  TWO  UNKNOWNS 


221 


EXERCISES 

1.  Graph  the  equation 

x2+y2+6x-16  =  0. 

2.  Graph  the  equation 

(z-7)2+(2/+8)2=36. 

228.  The  ellipse.    If  b=d  =  e  =  0,  the  general  qua- 
dratic, §  225,  reduces  to 

ax2+cy2+f=0.  (1) 

Show  that  the  equation  I6x2+25y2  =  400  is  of  this  form. 
Dividing  both  sides  of  this  equation  by  400, 

f!  .  J/!=1 
25"r16 

This  equation  is  of  the  form 

x2    v2 

Equation  (1)  is  better  adapted  to  computing  the  cor- 
responding values  of  x  and  y  than  equation  (2). 

x2     u2 
The  equation  o^+f^  =  1  maY  De  graphed  as  follows: 

1.  Solve  for  y, 


y=±-\/25-x2=±(.8)V25-x2. 

2.  Obtain  the  corresponding  values  of  x  and  y  as 
given  in  the  table,  Fig.  102. 


X 

y 

0 

±4 

1 

±3.9 

2 

±3.7 

3 

±3.2 

4 

±2.4 

5 

±0 

1 

t 

, 

?> 

P 

^ 

J 

« 

* 

7I 

( 

4, 

-' 

iIL.2. 

« 

u 

1/ 

V 

, 

J 

1(5'0J 

r 

J  V 

r 

<;4 

i 

'(1-1 

*( 

4) 

4- 

• 

* 

b 

(0 

# 

Jvi-)- 

Fig.  102 


222  THIRD-YEAR  MATHEMATICS 

3.  Graph  these  values  and  draw  the  curve  through 
the  points  thus  obtained. 

The  curve  in  Fig.  102  is  an  ellipse. 
Equation  (2)  readily  shows: 

1.  That  the  curve  is  symmetric  with  respect  to  the 
axes. 

2.  That  the  intercepts  on  the  axes  are  =  a  and  =*=  b. 
The  lengths  a  and  b  are  the  semiaxes  of  the  ellipse. 

If  a  =  b,  the  ellipse  reduces  to  a  circle 

EXERCISES 

1.  Graph  the  equation 

4x2+?/2  =  20. 

2.  Graph  the  equation 

x2+4?/2=16. 

229.  The  Hyperbola.     If  in  the  equation 
ax2+q/2+/=0, 

c  is  negative  and  a  positive,  the  equation  may  be  changed 
to  the  form 

a2    b2 

For  a  =  5  and  6  =  4,  this  reduces  to 

25     16 
The  equation  may  be  graphed  as  follows: 
1.  Solving  for  y, 

2/===b(.8)l/x2"^25. 


EQUATIONS  IN  TWO  UNKNOWNS 


223 


2.  The  corresponding  values  of  x  and  y  are  computed 
and  tabulated  as  in  Fig.  103. 


X 

y 

0 

*(-8) 

±5 

0 

±6 

±2.6 

±7 

±3.9 

±8 

±5. 

±9 

±6. 

; 

• 

( .  8)  y  —  25,  imaginary 


.... 

i 

' 

(-9, 

^,6) 

N 

L 

/ 

(- 

3T 

V 

rV" 

-finvp) 

X=k&61> 

jFi 

5.2.6) 

C 

T 

(- 

5. 

w 

>.( 

>J 

JT 

1 

? 

i 

k 

.v>  j 

v> 

s 

(-' 

'■  -3  I 

>y 

. 

p) 

(-9,- 

5) 

y 

Fig.  103 


3.  The  curve,  Fig.  103,  is  the  graph  of  the  equation 

16 
The  curve  is  called  a  hyperbola. 


25     16 


EXERCISES 

1.  Graph  the  equation 

z2-?/2-4  =  0. 

2.  Graph  the  equation 

9     4 

230.  The  graph  of  an  equation  of  the  form 

xy=c 

is  also  a  hyperbola.     This  equation  was  discussed  in 
§24. 


224 


THIRD-YEAR  MATHEMATICS 


If  c  =  8,  the  graph  xy  =  8,  Fig.  104,  is  easily  obtained 
by  means  of  the  values  in  the  table,  Fig.  104. 


.  x 

y 

1 

+8 

2 

+4 

3 

+2.7 

4 

+2 

5 

+  1.6 

6 

+  13 

7 

+11 

8 

+1 

-1 

-8 

-2 

-4 

-3 

-2.7 

-4 

-2 

-5 

-1.6 

-6 

-1.3 

-7 

-1.1 

-8 

-1 

1 

Tfl 

-| 

\ 

i 

V 

\ 

\ 

V 

s 

s 

N 

s 

- 

( 

S 

\ 

\ 

\ 

\ 

Fig.  104 


EXERCISES 


1.  Graph  the  equation 

2.  Graph  the  equation 


xi/  =  24. 
xy  =  5. 


231.  Two   straight  lines.     If  the  general  quadratic 
equation  is  the  product  of  two 
linear  factors,  the  graph  consists 
of  two  straight  lines,  as  illustrated 
in  the  following  example : 

z2+2:n/+2/2+2z+2i/-3  =  0. 
By  grouping  terms, 

(*+2/)2+2(z+2/)-3  =  0 
/.  (x+H-3)(z+2/-l)  =  0. 
The  graph  consists  of  two  parallel 
straight  lines,  Fig.  105.  Fig.  105 


sij:    x 

^  + 

_Vc 

S,_     V 

s^_   s^ 

N,    O-     ^»«-         -X 

s      Sr  _L 

$>.       S^. 

^£U  S^ 

4g»     S^. 

s*r  s^ 

%-    s^ 

S        S,. 

s       s 

^    - 

EQUATIONS  IN  TWO  UNKNOWNS  225 

Solution  of  Simultaneous  Quadratics 

232.  To  solve  a  system  of  quadratic  equations  in  two 

unknowns,  as 

ry-x2  =  2 

x  —  y2  =  5 

one  of  the  unknowns  may  be  eliminated  by  substitution. 

Since  y  =  2-\-x2,  the  second  equation  is  changed  to 
z-(2+x2)2  =  5, 
or  x—4:—4:X2  —  x4  =  5, 

.'.     x*+4:X2-x+9  =  0. 

This  is  an  equation  of  the  fourth  degree.  So  far 
the  student  knows  no  general  method  of  solving  an 
equation  of  the  fourth  degree  and  therefore  he  will  find 
it  impossible  to  solve  some  systems  of  simultaneous 
quadratics. 

A  similar  situation  was  found  in  the  study  of  factoring 
polynomials.  Not  being  able  to  work  out  a  general 
method  by  which  any  polynomial  may  be  factored,  we 
made  a  study  of  certain  typical  forms  of  polynomials. 
Methods  were  then  worked  out  for  factoring  such  special 
forms. 

Similarly  we  shall  now  study  only  certain  cases  of 
simultaneous  quadratics  and  find  the  proper  methods  of 
solution. 

233.  Case  I.  The  form  of  the  equations  is  such  that 
either  x  or  y  may  be  eliminated  by  addition  or  subtraction. 
The  following  example  illustrates  case  I: 

EXERCISES 

Solve  the  system  of  equations 

f  x2+  2/2=  25 
\4z2+9?/2=144 


226 


THIRD-YEAR  MATHEMATICS 


1.  Graphical  solution. — The  graph 
of  the  first  equation  is  a  circle,  §  226. 
The  graph  of  the  second  equation  is  an 
ellipse,  §  228. 

The  two  graphs  intersect  in  four 

points,   Fig.  106,   the   co-ordinates  of 

which    must    satisfy   both    equations. 

Hence     there    are    four    solutions: 


m 


s 


i) 


Fig.  106 
(4.1,  2.9),  (4.1,  -2.9),  (-4.1,  2.9),  and  (-4.1,  -2.9). 

2.  Algebraic  solution. — Multiplying  the  first  equation  of  the 
system  by  4  and  leaving  the  second  equation  as  it  is,  we  have 


and 

By  subtracting, 


4x2+4?/2=100 
4z2+9?/2=144 


5y2=  44 

.\  yi=vX8,  2/2= -VX8. 

Substituting  the  value  of  y\  in  the  first  of  the  given  equations, 

x2+8.8  =  25 
/.    z2=16.2 
/.    3==*^  16.2. 

Hence  we  have  the  two  following  solutions: 

(/lO,  v/878)  and  (-VTjO,  /O). 


Similarly,  by  substituting  ?/2=  —  ^8.8  in  the  first  of  the  given 
equations,  we  have  the  solutions: 

(VTO,  -/O)  and  (-/16^2,  -/O). 
Verify  the  four  solutions  by  means  of  the  graph,  Fig.  106. 


EQUATIONS  IN  TWO  UNKNOWNS 


227 


2. 


Solve  the  following  systems : 
5x2-2?/  =  30 

2z2+?/2  =  57 

f2x2+2/2-33  =  0 
\z2+2i/2-54  =  0 

x2+y*=lQ 
4z2-9i/2  =  36 

z2    x?/_8 
3     ~4  ~6 


15. 


6. 


7. 


J8. 


x2+y2=l 
x2  —  y2=l 

4x2-9?/2  =  36 
z2-f-4?/2  =  4 


(9x2+4?/2  =  36 
\9x2+162/  =  33 

x2_j_2/2=25 


-2_ 


3j/  =  21 


234.  Case  II.     One  equation  can  be  resolved  into  two 
linear  factors.     The  following  example  illustrates  case  II: 

Solve  the  following  system  of  equations: 

|4?/2-3z2+a;?/+15x-202/  =  0 
\  z2+  i/2  =  25 

1.  Graphical  solution. — By  factoring,  the  first  equation  is 
changed  to  the  form 

(4y-Sx)(x+y-5)=0. 

Hence  the  graph  consists  of  two  straight  lines,  Fig.  107. 

The  graph  of  x2+y2  =  25  is  a  circle, 
§226. 

•  .'.  the  co-ordinates  of  the  four 
points  of  intersection  give  the  solutions 
of  the  system. 

2.  Algebraic  solution. — According 
to  the  graph,  Fig.  107,  two  of  the  solu- 
tions of  the  system  are  obtained  from 
the  points  of  intersection,  A  and  B,  of 

the  circle  with  the  straight  line  whose  equation  is  x-{-y— 5  =  0. 
The  other  two  solutions  are  given  by  the  points  of  intersection, 
C  and  D,  of  the  circle  with  the  straight  line  whose  equation  is 
4y-&*-»0. 


c.            \>  \U?r 

t               ^SL 

&*          Nl. 

1&£            \ 

\'*y           i 

\y                    T 

jt  tlS                 ^ 

s             "">->-^ -^"^ 

Fig.  107 


228  THIRD-YEAR  MATHEMATICS 

This  suggests  that  the  solutions  of  the  original  system  be 
obtained  by  solving  the  systems: 

(x  +y  -5  =  0  (4?/-3x  =  0 

\      x2+y2  =  25  and  \  x2+y2  =  25 

Both  systems  should  be  solved  by  eliminating  one  of   the 
unknowns  by  substitution. 

Solving  the  first  system,  we  have: 

x  =  5— y. 

Substituting,  (5-?/)2+?/2  =  25, 

.'.  2y2-10y  =  0, 
•'•  2/i  =  0, 2/2  =  5. 

The  corresponding  values  of  x  are  found  by  substituting  these 
values  of  y  into  the  equation 

x  =  5— y. 

This  gives  the  solutions:   (5,  0)  and  (0,  5). 
Find  the  remaining  solutions. 


EXERCISES 

Solve  the  following  systems: 


,     (x2—5xy 

#    \x2-y2  = 

,    (x2-y2  =  0 
"  \x2+?/2  =  8 


5xy+6y2  =  0  .      fx2=y2 


27  +      \Sx2+5y2  =  32 

.       Ux2-9y2  =  0 
1  '  \4x2+9?/2-i  =  0 


x*+xy  =  0  .       (x2-3xy  =  0 

x2-xy+y2  =  27  *  *  \5x2+3?/2-9  =  0 

x2+2:n/+?/2+z+?/-30  =  0 
a:?/ =  15 

ft    jm2-{-n2  —  5m— 5n=  —  4 

Multiply  the  second  equation  by  2  and  add  to  the  first  equation. 


EQUATIONS  IN  TWO  UNKNOWNS 


229 


235.  Case  III.     One  equation  is  of  the  form  xy  =  c. 
The  following  example  illustrates  this  case : 

Solve  the  system 

(x2+y2*=5 
\xy=2 


1.  Graphical  solution. — 
The  graph  of  the  first  equation 
is  a  circle  whose  radius  is  2 . 2, 
approximately,  Fig.  108. 

The  graph  of  the  second 
equation  is  a  hyperbola. 

The  two  curves  intersect 
in  points  A,  B,  C,  and  D, 
which  determine  the  solutions. 

2.  Algebraic  solution. — 
Multiply  the  second  equation 
of  the  given  system  by  2  and 
add  the  resulting  equation  to 

the  first.    This  gives  the  equation 


Sj 

% 

\ 

^ 

»' 

s. 

. 

7 

\ 

s* 

* 

\ 

N\ 

/ 

SvJs 

\ 

■^ 

1 

\ 

\ 

\ 

3 

—1 

r 

7 

>\ 

\ 

q5& 

j 

\ 

-4 

f 

* 

\ 

^Y\ 

X-i 

\j 

s 

\ 

\ 

Fig.  108 


x2+2xy+y2  =  9, 

in  which  the  left  number  is  a  perfect  square. 

Extracting  the  square  root  of  both  members,  we  obtain  the 
two  linear  equations 

x+y  =  +S. 
and 

x+y=-S. 

The  graphs  of  these  two  equations  must  pass  through  the 
four  points  of  intersection  of  the  hyperbola  and  circle,  Fig.  108. 

Therefore  two  of  the  four  required  points  may  be' located  by 
graphing  the  straight  line  x -\-y  =  3  and  the  simpler  of  the  given 
equations,  xy  =  2. 

The  other  two  points  may  be  determined  by  the  graphs  of 
x+y=  —3  and  xy  =  2. 


230 


THIRD-YEAR  MATHEMATICS 


This  suggests  that,  instead  of  solving  the  original  system, 
we  may  solve  the  two  systems, 


(x+y  =  S 

Substituting  for  x  its  equal 

3-2/, 

(3-2/)2/  =  2, 

/.     ^-32/4-2  =  0. 

From    this    we    find    the 
solutions: 

(1,  2)  and  (2,  1). 


and 


z+?/=-3 
xy  =  2 


Substituting  for  x  its  equal 

-3-2/, 

(~3-2/)2/  =  2, 

2/2+32/+2  =  0. 

Show   that  the  solutions 
are: 

(-2,  -1)  and  (-1,  -2). 


Often,  when  neither  of  the  given  equations  is  of  the 
form  xy  =  c,  we  can  obtain  linear  equations  by  addition  or 
subtraction,  and  extraction  of  the  square  root  or  factoring. 

Thus,  in  exercise  2  below,  the  two  given  equations 
may  be  added.  The  resulting  equation  may  be  then 
divided  by  2.  The  two  linear  equations  are  obtained  by 
extracting  the  square  root  of  both  members  of  the  last 
equation. 

In  exercise  3  the  two  given  equations  may  be  added. 
By  factoring  both  members  of  the  resulting  equation,  we 
obtain  two  linear  equations. 


EXERCISES 

Solve  the  following  systems: 


fz2+2/2 
\xy-Q 


=  13 


2. 


>•{ 


(x2-5xy+y2=-2 
\z2+9a:2/+2/2  =  34 

x2+xy+y2  =  7 
x+xy+y  =  5 

r2-f-rs  =  5 
rs-f-s2  =  4 


5. 


7. 


8. 


z2+2/2+*+2/=18 
xy  =  Q 

(x2+xy  =  6 
*  W+2/2=10 

(t2+u2  =  5 
\tu-\-t+u  =  5 

2 


(r2-H 
\rs  =  ! 


EQUATIONS  IN  TWO  UNKNOWNS  231 

236.  Case  IV.  All  terms  containing  the  unknowns 
are  of  the  second  degree.  If  in  an  equation  all  terms  contain- 
ing the  unknowns  are  of  the  same  degree  the  equation  is 
homogeneous  with  respect  to  these  terms.  The  following 
example  illustrates  case  IV: 

Solve  the  following  system  of  equations:  % 

(x2+xy+y2  =  7 
\x2-xy+y2=l9 

Multiplying  the  first  equation  by  19,  the  second  by  7, 
19x2+19xy+19y2=19-7 
and  7x2-  7xij+  7y2=19>7. 

Subtracting  one  of  these  equations  from  the  other,  we  have 

12x2+2Gxy+  12y2=0 
Dividing  by  2,  Qx2+I3xy+  6y2  =  0. 

Factoring,  (2x+3y)(Zx+2y)  =  0. 

.'.    we  can  replace  the  given  system  by  the  following: 


(2x+3y  =  0              ,    (3x+2y  =  0 
\x2+xy+y2  =  7            \x2+xy+y2  =  7 

Solve  these  two  systems. 

EXERCISES 

Solve  the  following  systems: 

(x2+3xy-y2  =  3 
'  \2x2+5xy+y2  =  8 

(4a2-2xy  =  y2-l6 
'  \5x2-7xy+36  =  0 

(x2+xy  =  75 
'  \y2+x2=125 

(x2+3xy  =  7 
'  \x2-xy+y2  =  3 

(x2+2xy+2y2=10 
*   [3x2-xy-y2  =  51 

(x2-xy-\-y2  =  21 
'  \y2-2xy=-15 

Solution  of  Equations  of  Degree  Higher  than 
the  Second 

237.  One  equation  is  divisible  by  the  other.  Some 
systems  of  quadratic  equations  and  of  equations  of  higher 
degree  may  be  solved  by  dividing  one  equation  by  the 
other,  as  in  the  following  example: 


232 


THIRD-YEAR  MATHEMATICS 


Solve  the  system  of  equations: 
\x+y  =  3 


(1) 

(2) 


Since  x+y  is  a  non-zero  constant,  we  may  divide  the  first 
equation  by  the  second  without  losing  a  solution  of  the  given 
system. 

This  gives  y 

x*-xy+y*  =  S.  (3) 

Graphical  solution. — Solving  equation  (1)  for  y, 

y=f9-x\ 

By  means  of  the  table  of  cube  roots  verify  the  correspond- 
ing values  of  x  and  y,  as  given  in  table  (1),  Fig.  109. 


X 

y 

X 

y 

0 

2.1 

0 

*1.7 

1 

2 

1 

2,  -1 

2 

1 

2 

1,      1 

3 

-2.6 

3 

Imaginary 

4 

-3.8 

-1 

-1,  -2 

-1 

2.1 

-2 

-1,  -1 

-2 

2.6 

-3 

Imaginary 

-3 

3.3 
3.8 

-4 

(1) 

(3) 

\ 

\ 

\ 

\ 

< 

3 

^ 

L* 

•< 

^tt 

* 

V 

\ 

- 

_  j 

!    • 

'/ 

-i 

Q 

--1 

J  A 

.i 

\ 

, 

A 

\ 

k 

^ 

\ 

i 

S 

- 

4 

•> 

s 

Fig.  109 

Graph  equations  (1)  and  (2)  and  give  the  solution  of  the 
given  system. 

Solving  equation  (3), 


'.±Vl2-Sx2 


EQUATIONS  IN  TWO  UNKNOWNS  233 

By  means  of  this  equation  obtain  the  values  in  table  (3), 
Fig.  109. 

It  is  seen  that  the  graph  of  equation  (3)  passes  through  the 
points  of  intersection  of  the  graphs  of  equations  (1)  and  (2). 

Hence,  by  solving  the  system 

fx2-^+2/2  =  3 
1         x+y  =3, 

we  are  able  to  find  the  required  solutions  of  the  given  system. 
Give  the  complete  algebraic  solution. 

EXERCISES 

Solve  the  following  systems: 

,'a2-62  =  3  6    fs2-*2  =  228 


a-b  =  l  \st-t2  =  42 

,'a3+&3  =  18  6    jV3-s3  =  56 


a+6  =  6  \r2+rs+s2  =  28 

rx3+2/3==27  7    (a4+a2&2+64  =  91 

\x+y  =  S  V+a&+&2=13 

J  a26+a&2=  126  fa2+a&+262  =  74 

\a+6  =  9  \2a2+2a&+62  =  73 

Solution  of  Irrational  and  Fractional  Equations 

238.  Introduction  of  a  new  variable.  Some  equa- 
tions in  which  the  unknowns  appear  in  combinations  may 
be  simplified  by  using  a  new  symbol  in  place  of  these 
combinations.     Thus  the  equations 

(x+2/)2+3(z-h/)  =  -2,  x+y+V^+y^,  and^+^8 

may  be  written  respectively 

a2+3a=-2,   a2+a  =  6,  and  x2-f?/2  =  8. 

This  device  may  be  used  in  the  exercises  below 


234 


THIRD-YEAR  MATHEMATICS 


EXERCISES 

Solve  the  following  systems: 


f: 


a+b+V  a+b  =  § 


'2-B2=10 

Denote  V  a+b  by  x.    Solve  the  first  equation  for  x  and  obtain 
two  linear  equations  in  a  and  b. 


[1 

1   1 

X 

y  3 

1 

l    l 

X2 

y2   4 

D,l      1 

Put  -  =  a,  -i 


6. 


3*  {v*+fi=6 

Let  fx  =  a,  Vy  =  b. 


4. 


6. 


1 


X2/      I/2 


1+1  =  4 

r+s— r2s2 


r2_|_s2_r4s4  = 


fa6+5a+56  =  23 
\4a6+3a+36  =  2a6(a+6) 


MISCELLANEOUS   EXERCISES 

239.  Solve  the  following  problems : 

1.  The  perimeter  of  a  rectangle  is  22  inches.  If  the  cube  of 
its  length  is  added  to  the  cube  of  its  width  the  result  is  407. 
Find  the  area  of  the  rectangle. 

2.  The  difference  of  the  cubes  of  two  consecutive  numbers  is 
817.    Find  the  numbers.     (Chicago.) 

3.  The  sum  of  two  numbers  multiplied  by  the  greater  is 
126  and  their  difference  multiplied  by  the  less  is  20.  Find  the 
numbers.     (Princeton.) 

4.  The  area  of  a  rhombus  is  24  sq.  in.  and  the  sum  of  its 
diagonals  is  14  inches.     Find  the  length  of  one  side.     (Harvard.) 

|5.  The  sum  of  the  volumes  of  two  cubes  is  559  cu.  in.  and 
the  sum  of  their  lengths  is  13  inches.  What  is  the  height  of 
each  cube  ? 


EQUATIONS  IN  TWO  UNKNOWNS  235 

6.  At  his  usual  rate  a  man  can  row  15  mi.  downstream  in  5  hr. 
less  time  than  it  takes  him  to  return.  Could  he  double  his  rate, 
his  time  downstream  would  be  2  hr.  less  than  his  time  upstream. 
What  is  his  usual  rate  in  still  water  and  what  is  the  rate  of  the 
current  ?     (Board.) 

J7.  The  diagonal  of  a  rectangle  is  13  ft.  long.  If  each  side 
were  longer  by  2  ft.,  the  area  would  be  increased  by  38  square 
feet.    Find  the  length  of  the  sides. 

8.  Two  men,  A  and  B,  start  at  the  same  time  from  a  certain 
point  and  walk  east  and  south  respectively.  At  the  end  of  5  hr. 
A  has  walked  5  mi.  farther  than  B,  and  they  are  25  mi.  apart. 
Find  the  rate  of  each. 

9.  If  a  number  of  two  digits  is  divided  by  the  sum  of  the 
digits,  the  quotient  is  2  and  the  remainder  is  2.  If  it  is  multi- 
plied by  the  sum  of  the  digits,  the  product  is  112.  Find  the 
number.     (Board.) 

10.  Three  men,  A,  B,  and  C,  can  do  a  piece  of  work  together 
in  1  hr.  and  20  minutes.  To  do  the  work  alone  C  would  take 
twice  as  long  as  A  and  2  hr.  longer  than  B.  How  long  would 
it  take  each  to  do  the  work  alone  ?     (Board.) 

11.  Find  two  numbers  such  that  their  sum,  difference,  and 
the  sum  of  their  squares  are  in  the  ratio  5:3:51.     (Yale.) 

12.  The  sum  of  the  ages  of  a  father  and  his  son  is  100  years, 
and  one-tenth  of  the  product  of  the  numbers  of  years  in  their 
ages  minus  180  equals  the  number  of  years  in  the  father's  age. 
What  is  the  age  of  each  ? 

13.  Solve  the  equations 

l  =  a+(n-l)d 
2s  =  n(a+l) 

taking  I  and  n  as  the  two  unknown  numbers.    Find  I  and  n  when 

113 

a  =  -,  d  =  -—,s=--.     (Princeton.) 

14.  Two  men,  A  and  B,  dig  a  trench  in  20  days.  It  would 
take  A  alone  9  days  longer  to  dig  it  than  it  would  B.  How  long 
would  it  take  A  and  B  each  working  alone  ?     (Yale.) 


236  THIRD-YEAR  MATHEMATICS 

15.  Two  automobiles  run  336  miles.  The  winning  car  wins 
by  4  hr.  by  going  2  mi.  an  hour  faster  than  the  other.  What  was 
the  winner's  time  and  speed  ?     (Sheffield.) 

16.  Two  men  work  on  a  job  and  each  receives  36  dollars. 
One  of  them,  however,  has  worked  2  days  less  than  the  other 
and  is  paid  20  cents  more  a  day.  Find  his  daily  wages  and  the 
number  of  days  he  worked.     (Sheffield.) 

17.  An  audience  of  360  persons  is  seated  in  rows  each  con- 
taining the  same  number  of  people.  They  might  have  been 
seated  in  four  rows  less  if  each  row  contained  3  more  people. 
How  many  rows  were  there  ?     (Board.) 

18.  Find  the  sides  of  a  rectangle  whose  area  is  unchanged 
if  its  length  is  increased  by  4  ft.  and  its  breadth  decreased  by  3  ft., 
but  which  loses  one-third  of  its  area  if  the  length  is  increased  by 
16  ft.  and  the  breadth  decreased  by  10  feet.     (M.I.T.) 

Solve  the  following  systems: 

(x*-Sxy-y>  =  9 
'  \2x2+2xy+3y*  =  7    (Board) 

(s*+st+s-t=-2 
•  \2s2-st-P  =  0 

(x+y  =  5-xy 

21-  L+2/=-        (Yale) 
I  xy 

22    (x>+xy+y*=133 

'  \x—Vxy+y  =  l 


23. 


(dxy  =  l 

|36:c2-r-  180xy+S6y2  =  -  35    (Sheffield) 


EQUATIONS  IN  TWO  UNKNOWNS  237 

(x2-4xy+4if-x+2y-G  =  0 
'  \±x2+12xy+9y2+2x+Zy-12  =  0 


27. 


x+-  =  l 

y 

2/+- =  4     (Harvard) 
a; 


»{; 


°2+st+P=l3s 
'«*-«H-£*7«     (Chicago) 

;5xY-2  =  3^ 


^+5?/=l     (Princeton) 

30.  J^-^  =  2   • 

\(i/a;->/y)(i/^)=30    (Yale) 

81    (2x+-Sx~y=10 
'  {Sy-2Vxy=-l 

(xy  =  80 
32.  jl_l  =  l 

[x    y    5 

33    fzy+28x?/-480=Q  34    Jz2+:n/+</2=2] 


\2s+y  =  ll     (Yale)  '  \x-^xy+y  =  S 

Summary 

240.  The  chapter  has  taught  how  to  graph  the  follow- 
ing quadratic  equations : 

1.  The   equation   y  =  ax2+bx+c  representing   a  pa- 
rabola. 

2.  The  equations  x2+y2  =  a2  and  x2+y2+ax+by+c  =  0 
representing  circles. 

3.  The  graph  of  ^2+|-2  =  l,  or  b2x2+a2y2  =  a2b2  repre- 
senting an  ellipse. 

x2     II2 

4.  The  equations  xy  =  c  and  ^—^  =  1   representing 
hyperbolas. 


238  THIRD-YEAR  MATHEMATICS 

241.  Outline  the  methods  of  solving  the  following 
systems  of  equations : 

x+y— 2  =  0  -    \x2-\-y2+xy  =  k 


4:X2-2x+5  =  y  '   [x2+2y2-xy  =  7 

(x2+y2  =  25  (^-2/3  =  21 

\4x2+9y2=U4:  \x-y  =  Z 

(x2+2xy+y2+x+y-12  =  0 
*'  \x2+y2  =  ZQ     ' 

(x2+y2  =  4:9  (x+y-x2y2  =  2 

\xy  =  24:  \x2+y2-xiy4  =  4: 


CHAPTER  XII 
AREAS  OF  SURFACES 

242.  In  the  world  about  us  some  forms  of  material 
objects  occur  frequently,  e.g.,  the  prismatic  forms  of 
boxes,  houses,  and  posts ;  the  cylindrical  forms  of  boilers, 
pipes,  columns,  and  tanks;  the  conical  forms  of  funnels, 
pails,  and  spires ;  and  the  spherical  forms  of  globes,  domes, 
and  balls.  It  is  the  purpose  of  this  chapter  to  study  the 
surfaces  of  these  forms  and  to  develop  formulas  by  means 
of  which  we  may  compute  the  areas  of  these  surfaces. 

Polyedrons.    Cylinders.    Cones 

243.  Polyedron.  A  geometrical  solid  bounded  en- 
tirely by  planes  is  a  polyedron.* 

EXERCISES 

1.  Give  some  examples  of  polyedrons. 

2.  What  is  the  least  number  of  planes  necessary  to  form  a 
polyedron  ? 

244.  Face.  Surface.  Edge.  Vertices.  The  bound- 
ary planes  of  the  polyedron  are  the  faces;  the  sum  of  the 
faces  is  the  surface ;  the  intersections  of  the  faces  are  the 
edges;  the  intersections  of  the  edges  are  the  vertices. 

245.  Classification  of  polyedrons.  Polyedrons  may 
be  classified  according  to  the  number  of  faces.     Thus, 

*  Some  authors  write  polyhedron  instead  of  polyedron. 
239 


240  THIRD-YEAR  MATHEMATICS 

Fig.  110,  a  polyedron  of  four  faces  is  a  tetraedron;  of  six 


Tetraedron  Hexaedron  Octaedron 

Fig.  110 


Dodecaedron 


Icosaedron 


faces  a  hexaedron;   of  eight  faces  an  octaedron;  of  fauefoe 
faces  a  dodecaedron;   of  twenty  faces  an  icosaedron 

246.  Relation  between  the  number  of  faces,  vertices, 
and  edges  of  a  convex  polyedron.  Count  the  number  of 
faces,  vertices,  and  edges  in  each  of  the  figures  in  §  245 
and  tabulate  the  results  as  below: 


Number  of  faces .  . 

4 

6 

8 

12 

Number  of  vertices 

4 

8 

6 

20 

Sum ! 

8 

14 

14 

32 

Number  of  edges 

6 

12 

12 

30 

Denoting  the  num- 
ber of  faces,  vertices, 
and  edges  by/,  v,  and  e, 
respectively,  compare  e 
with  f+v.  State  the 
relation  between  e  and 
f+v  in  the  form  of  an 
equation. 

The  mathematician 
LeonardEuler  (1707-83) 
proved  that  e+2=f+v. 

247.  Pyramid.  If  a 
line,  AB,  Fig.  Ill,  pass- 


Fig.  Ill 


AREAS  OF  SURFACES 


241 


ing  through  a  fixed  point,  A,  moves  always  touching  a 
convex  polygon,  CDEF,  whose  plane  does  not  contain  A,  it 
generates  a  pyramidal  surface.  The  generating  line  AB 
is  called  the  generatrix,  the  polygon  CDEF  the  directrix, 
and  the  fixed  point  A  the  vertex. 

The  solid  A'-C'D'E'F'  bounded  by  the  pyramidal 
surface  and  by  the  plane  of  the  polygon  is  a  pyramid. 

The  polygon  C'D'E'F'  is  the  base. 

The  portion  of  the  surface  between  the  vertex  and  the 
base  is  the  lateral  surface. 

The  perpendicular  distance  A'B'  from  the  vertex  to  the 
plane  of  the  base  is  the  altitude. 

The  edges  A'C ,  A'D',  etc.,  are  the  lateral  edges  of  the 
pyramid. 

248.  Cone.  If  aline,  Fig.  112,  passing  through  a  fixed 
point,  moves  always  touching  a  convex  closed  curve  whose 
plane  does  not  contain  the 
fixed  point,  it  generates  a  sur- 
face called  a  conical  surface. 
The  generating  line  is  the 
generatrix,  the  curve  the  direc- 
trix, and  the  fixed  point  the 
vertex.  The  generatrix  in  any 
position  is  called  an  element. 

The  solid  bounded  by  the 
conical  surface  and  the  plane 
of  the  curve  is  a  cone. 

The  curve  cut  from  the 
plane  by  the  conical  surface  is  the  base. 

The  portion  of  the  surface  between  the  vertex  and  the 
base  is  the  lateral  surface. 

The  perpendicular  distance  from  the  vertex  to  the 
plane  of  the  base  is  the  altitude. 


Fig.  112 


242 


THIRD-YEAR  MATHEMATICS 


The  two  parts  of  the  pyramidal,  or  conical,  surface 
on  opposite  sides  of  the  fixed  points  are  called  the  nappes. 

249.  Classification  of  pyramids.  Pyramids  are  tri- 
angular, quadrangular,  pentagonal,  etc.,  according  as  the 
bases  are  triangles,  quadrilaterals,  pentagons,  etc.,  Fig.  113. 


Triangular 
Pyramid 
Telraedron 


Quadrangular 
Pyramid 

Fig.  113 


Pentagonal 
Pyramid 


A  triangular  pyramid  is  a  tetraedron,  Fig.  113.  Any 
one  of  the  four  triangular  faces  of  a  tetraedron  may  be 
taken  as  the  base. 

If  the  base  of  a  pyramid  is  a  regular 
polygon,  Fig.  114,  and  if  the  altitude,  VH, 
meets  the  base  at  the  center  of  the  circum- 
scribed circle,  the  pyramid  is  said  to  be 
regular. 

The  altitude,  VA,  of  one  of  the  .tri- 
angular faces  of  a  regular  pyramid  is  the 
slant  height. 

EXERCISES 


Fig.  114 


1.  Give  examples  of  material  objects  of  the  form  of  a 
pyramid. 

2.  Prove  that  the  lateral  edges  of  a  regular  pyramid  are  equal. 

3.  Prove  that  the  lateral  faces  of  a  regular  pyramid  are  con- 
gruent isosceles  triangles. 

4.  Prove  that  the  slant  height  of  a  regular  pyramid  is  the 
same  for  all  lateral  faces. 


AREAS  OF  SURFACES 


243 


5.  The  lines  joining  the  midpoints  of  four  edges  of  a  tetrae- 
dron,  no  three  of  which  pass  through  the  same  vertex,  form  a 
parallelogram. 

6.  Prove  that  the  lines  connecting  the  middle  points  of  the 
opposite  edges  of  a  tetraedron  bisect  one  another.     (Harvard.) 

250.  Classification  of  cones.  A  cone  whose  base  is  a 
circle  is  a  circular  cone,  Fig.  115.  Only  circular  cones  are 
studied  in  this  chapter.  The  line  joining  the  vertex,  V, 
to  the  center,  C,  of  the  base  is  the  axis  of  the  cone.  A 
circular  cone  whose  axis  is  perpendicular  to  the  base  is  a 
right  circular  cone.  If  the  axis  is  oblique  to  the  base,  the 
v  v 


Fig.  115 


Fig.  116 


Fig.  117 


cone  is  said  to  be  oblique,  Fig.  116.  A  right  circular  cone 
may  be  generated  by  revolving  a  right  triangle  about  one 
of  its  sides  as  an  axis,  Fig.  117.  Hence  it  is  also  called  a 
cone  of  revolution.  The  distance  from  the  vertex  to  a 
point  of  the  base  of  a  cone  of  revolution  is  called  the  slant 
height. 

EXERCISES 

1.  The  altitude  of  a  right  circular  cone  is 
10  in.  and  the  radius  of  the  base  is  4  inches. 
Find  the  slant  height. 

2.  The  slant  height  of  a  right  circular  cone 
is  50  in.  and  is  twice  as  long  as  the  radius 
of  the  base.    Find  the  altitude. 

251.  Cylinder.  If  a  straight  line,  AB,  Fig.  118,  so 
moves  as  to  remain  parallel  to  a  fixed  straight  line  and  to 
touch  a  curve,  CD,  whose  plane  does  not  pass  through  the 


Fig.  118 


244 


THIRD-YEAR  MATHEMATICS 


fixed  line,  the  moving  straight  line  generates  a  cylindri- 
cal surface.     The  moving  line,  AB,   is  the 
generatrix.    The  generatrix  in  any  one  of  its  A  { 
positions  is  an  element  of   the  surface.     A 
solid  bounded  by  a  cylindrical  surface  and 
two  parallel  planes  is  a  cylinder,  Fig.  119. 
The  cylindrical  surface  is  the  lateral  surface  of  a'| 
the  cylinder,  the  two  plane  curves,  AB  and 
A'B',  are  the  bases. 


Fig.  119 


252.  Classification  of  cylinders.  If  the  base  is  a  circle 
and  if  the  elements  are  perpendicular  to  the  plane  of  the 
circle,   the   cylinder  is  called  a 

right  circular  cylinder,  Fig.  120. 
A  cylinder  whose  elements  are 
oblique  to  the  base  is  an  oblique 
cylinder,  Fig.  121.  A  right 
circular  cylinder  may  be  gen- 
erated by  revolving  a  rectangle, 
as  ace' a',  Fig.  120,  about  one 
of  the  sides.  Hence  it  is  also  FlG-  120 
called  a  cylinder  of  revolution. 

Give  examples  of  objects  of  cylindrical  form. 

253.  Prism.  A  prismatic  surface, 
Fig.  122,  may  be 
generated  by  mov- 
ing a  line  parallel 
to  itself,  so  that  it 
always  touches  a 
given  convex  poly- 
gon, as  ABCDE. 
A  solid  bounded  by 
parallel  planes  and 
a  prismatic  surface  is  a  prism,  Fig.  123 


Fig.  121 


E' 


c 
Fig.  123 


The  terms  base, 


AREAS  OF  SURFACES 


245 


lateral  surface,  and  altitude  have  the  same  meaning  for  the 
prism  as  for  the  cylinder. 

254.  Classification  of 
prisms.  A  prism  is  right, 
Fig.  124,  or  oblique,  Fig. 
125,  according  as  the  lateral 
edges  are  perpendicular  or 
oblique  to  the  planes  of  the 


s\ 

1 

1 

Fig.  124 


Fig.  125 


A  prism  is  said  to  be  triangular,  quadrangular, 
etc.,  according  as  the  base  is  a  triangle,  quadrilateral,  etc. 

EXERCISES 

1.  Prove  that  the  lateral  edges  of  a  prism  are  equal. 

2.  Prove  that  the  lateral  edges  of  a  right  prism  are  equal  to  the 
altitude. 

3.  Prove  that  the  lateral 
faces  of  a  prism  are  parallelo- 
grams. 

4.  What  is  the  locus  of  a 
straight  line  parallel  to  a  given 
straight  line  and  at  a  given 
distance  from  it  ? 

6.  What  is  the  locus  of  a 
point  having  a  given  distance 
from  a  given  straight  line  ? 

6.  Prove  that  a  lateral  edge 
of  a  prism  is  parallel  to  every  lateral  face  not  containing  it. 

Sections  Made  by  a  Plane 

255.  Section.  The  intersection  of  a  plane  with  the 
surface  of  a  solid  is  called  a  section,  as  ABODE,  Fig.  126. 

256.  Right  section.  If  the  plane  of  a  section  is  at 
right  angles  to  the  lateral  edges  of  a  prism  or  to  the  ele- 
ments of  a  cylinder,  it  is  a  right  section, 


246 


THIRD-YEAR  MATHEMATICS 


257.  Theorem:   The  sections  of  a  prism  made  by  paral- 
lel planes  cutting  all  the  lateral  edges  are  congruent. 

Given  the  prism,  AB,  Fig.   127,  plane  CF  ||  plane 
CF';  sections  CDEFG  and  C'D'E'F'G'. 

To  prove  that  ^»B 

CDEFG^C'D'E'F'G'. 

Proof:  Prove 

CD  ||  CD',  DE  ||  D'E',  etc. 
Prove 

CD  =  CD',DE  =  D'E',  etc. 
Prove 
ZCDE  =  ZC'D'E',  ZDEF  =  ZD'E'F', etc.  (§545). 

Hence 
CDEFG  2  C'D'E'F'G'.        Why  ? 


Fig.  127 


EXERCISES 

Prove  the  following: 

1.  The  right  sections  of  a  prism  are 
co  ngruent. 

2.  A  section  of  a  prism  parallel  to  the 
base  is  congruent  to  the  base. 

3.  The  bases  of  a  prism  are  congruent. 

4.  A  section  of  a  prism  made  by  a  plane 
parallel  to  a  lateral  edge  is  a  parallelogram. 


Fig.  128 


258.  Theorem:  A  section  of  a  cylinder  made  by  a  plane 
passing  through  an  element  is  a  parallelogram. 

Given  the  cylinder  AB}  Fig.  128;  the  element  DB; 
and  plane  F  passing  through  DB. 

To  prove  that  the  intersection  of  P  with  the  surface 
of  the  cylinder  AB  is  a  parallelogram. 


AREAS  OF  SURFACES  247 

Proof:  Plane  P  intersects  the  base  in  the  straight 
line  AD.    Why? 

Through  A  draw  line  AC  parallel  to  DB. 

AC  must  lie  in  plane  P.     Why  ? 

AC  is  also  an  element.    Why  ? 

Therefore  AC  lies  in  the  cylindrical  surface.     Why  ? 

Hence  AC  must  be  the  intersection  of  P  with  the 
cylindrical  surface.    Why  ? 

Draw  CB  and  show  that  it  is  the  intersection  of  P  with 
the  upper  base  of  the  cylinder. 

Show  that  ADBC  is  a  parallelogram. 

EXERCISE 

Show  that  ADBC,  Fig.  128,  is  a  rectangle  if  AB  is  a  right 
cylinder. 

259.  Theorem :  Sections  of  a  cylinder  made  by  parallel 
planes  cutting  all  elements  are  congruent. 

Given  the  cylinder  AB,  Fig.  129;  section  CD  ||  section 
CD'. 

To  prove  that  CD^CD'. 

Proof:  Let  E'  and  ¥'  be  two  fixed 
points  on  CD'  and  let  K[  be  any 
other  point  on  CD' . 

Draw  the  elements  through  E' ,  F', 
and  K',  meeting  CD  in  E,  F,  and  K 
respectively. 

Draw  triangles  EFK  and  E'F'K'. 

Prove  AEFK^AE'F'K'  (s.s.s.). 

Imagine  CD  placed  on  CD'  with  E  coinciding  with 
E'  and  F  with  F'. 

Then  K  must  fall  on  K'.    Why  ? 

Hence  every  point  on  CD  can  be  made  to  coincide  with 
the  corresponding  point  on  CD\  and  CD  %  CD'. 


248 


THIRD-YEAR  MATHEMATICS 


EXERCISES 

Show  that  the  following  statements  are  special  cases  of  the 
theorem  just  proved: 

1.  Sections  of  a  cylinder  parallel  to  the  base  are  congruent 
to  the  base. 

2.  The  bases  of  a  cylinder  are  congruent. 

3.  All  right  sections  of  a  cylinder  are  congruent. 

260.  Theorem:  If  a  pyramid  is  cut  by  a  plane  parallel 
to  the  base: 

1.  The  edges  and  altitude  are  divided  proportionally. 

2.  The  section  is  a 
polygon  similar  to  the  base. 

3.  The  areas  of  the 
section  and  the  base  are 
proportional  to  the  squares 
of  the  distances  from  the 
vertex. 

Given    the    pyramid 

V-A' B' CD' E', Fig.  130, 

and  plane  Q  ||  plane  P; 

VO'±P. 

VA 

1.  To  prove  that  yA'~VB'~VC 

Proof:  Through  V  draw  plane  R  \\  planes  P  and  Q. 
Show  that  the  conclusion  follows  (see  §  524). 

2.  To  prove  that  ABCDE^A'B'CD'E'. 
Proof:  Show  that  A  B  \\  A'B';   BC\B'C,  etc. 

.-.  AVABc*  AVA'B';  AVBC«>  AVB'C, 
etc. 

VB      AB       VB      BC     w,     9 

Why? 


A'B" 

BC 
A'B'~B'C. 


VB' 
AB 


VB'    B'C 
Why? 


AREAS  OF  SURFACES 


249 


Similarly  show  that 
BC 


CD 


7,  etc. 


etc. 


B'C    CD' 
Prove   that    ZABC=  LA' B'C ';     LBCD=  ZB'C'D' \ 

.♦.  ABCDE co  A'B'C'D'E'.    Why  ? 


3.  To  prove  that 


Proof: 


ABCDE       VO2 


A'B'C'D'E' 

VO'2' 

ABCDE 
A'B'C'D'E' 

AB2 
AJB'2' 

Why? 

AB2 

VB2 

VO2 

A'B,Z 

VB'2 

VO'2' 

ABCDE 
A'B'C'D'E' 

VO2 
VO'2' 

Why? 

Why? 


261.  Theorem:  A  section  of  a  cone  made  by  a  plane 
passing  through  the  vertex  is  a  triangle. 

Given  the  cone  V-AB  cut  by 
plane  P,  Fig.  131. 

To  prove  that  the  section  is  a 
triangle. 

Proof:    Plane  P  intersects  the 
base  AC B  in  the  straight  line,  CD. 

Draw    the    straight    lines    VC 
and  VD. 

Then  VC  and  VD  are  elements. 
Why? 

Therefore  VC  and  VD  lie  in  the  conical  surface. 
Why? 

But  they  also  lie  in  plane  P.    Why  ? 

Hence  the  straight  lines  VC  and  VD  are  the  inter- 
sections of  P  with  the  conical  surface,  and  the  section  is  a 
triangle.    Why? 


Fig.  131 


250 


THIRD-YEAR  MATHEMATICS 


262.  Theorem:  A  section  of  a  circular  cone  made  by  a 
plane  parallel  to  the 
base  is  a  circle. 

Given  the  circu- 
lar cone  V-A'B', 
Fig.  132,  and  plane 
Q  ||  plane  P. 

To  prove  that 
the  section  ACDB 
is  a  circle. 

Proof:  Take 
any  two  points,  C 
and  D,  on  A  B. 

Draw  the  elements  VCC  and  VDD*. 

Draw  VO'  intersecting  the  plane  of  AB  in  0. 

Draw  CO,  DO,  CO',  D'O'. 

Prove  that  A VOC oo  A VO'C;  &VOD™  AVO'D'. 
VO  =  CO 
VO'    CO'' 
VO      DO 
VO'    DV 

■      CO^=DO_      Why? 
CO'    DV  y 

Since  C'O'^D'O',  it  follows  that  CO  =  DO. 

.'.  the  section  ACDB  is  a  circle.     Why  ? 

EXERCISES 

1.  The  area  of  the  base  of  a  pyramid  is  1 10  square  feet.  The 
area  of  the  section  of  the  pyramid  parallel  to  the  base  and 
5  ft.  from  it  is  80  square  feet.  Find  the  altitude  to  two 
decimal  places. 

2.  The  base  of  a  pyramid  is  50  sq.  in.  and  the  altitude  6 
inches.  How  far  from  the  vertex  must  a  plane  be  passed  that 
the  area  of  the  section  may  be  half  as  large  as  the  area  of 
the  base  ? 


Then 


Why? 
Why? 


AREAS  OF  SURFACES 


251 


EXERCISES 

1.  Show  that  the  axis  of  a  right  circular  cone  passes  through 
the  center  of  every  section  parallel  to  the  base. 

2.  Prove  that  the  radius  of  the  section  of  a  circular  cone  made 
by  a  plane  parallel  to  the  base,  and  the  radius  of  the  base  are 
proportional  to  the  distances  from  the  vertex  to  the  cutting 
plane  and  to  the  plane  of  the  base. 

263.  Parallelopipeds.  A  prism  whose  bases  are  paral- 
lelograms is  a  parallelopiped,  Figs.  133  to  136.    A  paral- 


Parallelopiped 


Fig.  133 


M 


^  \ 


^  ^    \ 


\LA 


Right 
Parallelopiped 


Rectangular 
Parallelopiped 


Fig.  134    '         Fig.  135 


Cube 


Fig.  136 


lelopiped  whose  lateral  edges  are  perpendicular  to  the 
bases  is  a  right  parallelopiped,  Fig.  134.  A  right  parallel- 
opiped whose  bases  are  rectangles  is  a  rectangular  paral- 
lelopiped, Fig.  135.  A  parallelopiped  all  of  whose  faces 
are  squares  is  a  cube,  Fig.  136. 


EXERCISES 

1.  State  the  difference  between  a  right  parallelopiped  and 
a  rectangular  parallelopiped. 

2.  Show  that  the  faces  of  a  rectangular  parallelopiped  are 
all  rectangles. 

3.  Prove  that  the  opposite  faces  of  a  parallelopiped  are 
parallel. 

Use  §  545. 

4.  Prove  that  a  section  of  a  parallelopiped  made  by  a  plane 
cutting  four  parallel  edges  is  a  parallelogram. 

5.  Prove  that  the  diagonals  of  a  cube  are  equal. 


252 


THIRD-YEAR  MATHEMATICS 


6.  Find  the  diagonal  of  a  cube  whose  edge  is  2;  3.4;  e. 

7.  Prove  that  the  square  of  a  diagonal  of  a  rectangular 
parallelopiped  is  equal  to  the  sum  of  the  squares  of  three  edges 
meeting  in  the  same  vertex. 

8.  Find  the  diagonal  of  a  rectangular  parallelopiped  whose 
edges  are  6,  8,  and  10  respectively. 

9.  Prove  that  the  diagonals  of  a  rectangular  parallelopiped 
are  equal  and  bisect  each  other. 

10.  Find  the  length  of  the  diagonal  of  a  rectangular  paral- 
lelopiped whose  edges  from  any  vertex  are  4,  6,  and  8. 

11.  Find  the  edge  of  a  cube  whose 
diagonal  is  12  inches. 

264.  Truncated  prism.  A  portion  of 
a  prism  included  between  the  plane  of 
the  base  and  the  plane  of  a  section  not 
parallel  to  the  base  is  a  truncated  prism, 
Fig.  137. 

265.  Frustum  of  a  pyramid.  Altitude. 
The  portion  of  a  pyramid  included  be- 
tween the  plane  of  the  base  and  the  plane 
of  a  section  parallel  to  the  base  is  a  frustum 
of  a  pyramid,  Fig.  138.  The  perpen- 
dicular, 00',  intercepted  between  the 
planes  of  the  bases  is  called  the  altitude 
of  the  frustum. 

EXERCISES 

1.  Show  that  the  lateral  faces  of  a  frustum  of  a  pyramid  are 
trapezoids. 

2.  Show  that  the  lateral  faces  of  a  frustum  of  a  regular 
pyramid  are  congruent  trapezoids. 

3.  Prove  that  a  plane  bisecting  the  altitude  and  parallel  to 
the  plane  of  the  bases  of  a  frustum  of  a  pyramid  forms  a 
section  whose  perimeter  is  half  the  sum  of  the  perimeters  of 
the  bases. 


Fig.  137 


Fig.  138 


AREAS  OF  SURFACES 


253 


Fig.  141 


266.  Slant  height  of  a  frustum.  The  altitude,  AB, 
Fig.  139,  of  a  lateral  face  of  a  frustum  of  a  regular  pyramid 
is  the  slant  height  of  the  frustum. 

267.  Frustum  of  a  cone. 
The  portion  of  a  cone  included 
between  the  plane  of  the  base 
and  the  plane  of  a  section  parallel 
to  the  base  is  a  frustum  of  a  cone, 
Fig.  140. 

268.  Sections  of  a  cone.  In 
the  discussion  of  the  plane  sec- 
tions of  a  right  circular  cone 
the  following  cases  may  be 
considered : 

Let  P  be  a  plane  perpen- 
dicular to  plane  AVB,  Fig.  141. 

1.  If  P,  Fig.  141,  passes 
through  the  vertex  V  and  an 
element  VD,  it  cuts  the  surface 
in  two  intersecting  straight  lines, 
as  DD'  and  CC. 

2.  If  P  does  not  pass  through 
the  vertex  V,  and  if  it  is  per- 
pendicular to  the 
axis  VC,  the  sec- 
tion is  a  circle, 
Fig.  142. 

3.  If  P,  Fig. 
143,  is  not  per- 
pendicular to  the 
axis,  but  meets 
both  of  the  elements  VA  and  VB,  the  section  is  an  ellipse. 

4.  If  P,  Fig.  144,  is  parallel  to  one  of  the  elements, 
the  section  is  a  parabola. 


Fig.  143 


Fig.  144 


254 


THIRD-YEAR  MATHEMATICS 


5.  If  plane  P,  Fig.  145,  meets  some  of  the  elements 
produced,  the  section  is  a  hyperbola. 


Fig.  145 
Thus  we  have  the  following  sections  of  a  cone: 


C       U)     D 

(2) 

(8) 

U) 

Two  intersecting 

Circle 

Ellipse 

Parabola 

straight  lines 

Hyperbola 
Fig.  146 


It  was  seen  in  chapter  XI  that  these  curves  represent 
graphically  quadratic  equations  in  two  unknowns. 

A  very  extensive  study  of  these  sections  is  made  in 
analytic  geometry. 


AREAS  OF  SURFACES 


255 


Areas 

269.  Theorem:    The  lateral  area  of  a  prism  is  equal  to 

the  perimeter  of  a  right  section  multiplied  by  the  lateral  edge. 

In  symbols  this  may  be  expressed  by  the  equation 

L=p  •  e, 

L  denoting  the  lateral  area,  p  the  perimeter  of  the  right 
section,  and  e  the  length  of  a  lateral  edge. 

Given  the  prism  AD';  the  right 
section  FK,  Fig.  147. 

To  prove  that        L  =  p  •  e. 

Proof:  Show  that  the  lateral  edges 
are  equal. 

Show  that  the  lateral  faces  are 
parallelograms. 

Show  that  the  sides  of  the  section 
FK  are  the  altitudes  of  these  parallelo- 
grams. 

Hence      AB'  =  FG  -  BW=FG  •  e, 

BC'=GR  •  CC7=GH  -e,  etc. 

Adding,  AB'+BC'+  etc.  =  (FG+GH+eic.)e, 
or  L=p  •  e. 


EXERCISES 

1.  Prove  that  the  lateral  area  of  a  right  prism  is  equal  to  the 
perimeter  of  the  base  by  the  altitude. 

2.  Find  the  lateral  area  of  a  prism  whose  lateral  edge  is 
18  cm.  and  whose  right  section  has  a  perimeter  equal  to  29 
centimeters. 


3.  Find  L  (1)  if  e  =  2.75,  p  =  5.26 
(2)ife  =  5i,p  =  10f  . 
(3)  if  6  =  12.14.^  =  25^ 


256 


THIRD-YEAR  MATHEMATICS 


4.  Find  p   (1)  if  L  =  20.26,  e  =  12.48 
(2)  if  L=19f,  e=  6.92 

6.  Find  the  lateral  area  of  a  column  having  the  form  of  a 
regular  hexagonal  right  prism,  if  one  side  of  the  base  is  5  ft., 
and  if  the  altitude  is  8  feet. 

6.  Find  the  total  surface  of  a  cube  whose  edge  is  2. 6  centi- 
meters. 

7.  Find  the  total  area  of  a  right  triangular  prism,  if  the 
base  is  an  equilateral  triangle  whose  side  a =2. 7  in.  and  if  the 
altitude  h = 8 . 4  inches. 

8.  How  many  square  inches  of  copper  lining  will  be  required 
to  line  the  sides  and  base  of  a  tank  9  in.  high,  9f  in.  wide,  and 
20  in.  long  ? 

9.  How  many  square  feet  of  lead  will  be  required  to  line  a 
rectangular  cistern  9 J  ft.  long,  7  ft.  wide,  and  5  ft.  deep  ? 

270.  Theorem:  The  lateral  area  of  a  regular  pyramid  is 
equal  to  one-half  the  product  of  the  slant  height  by  the  perim- 
eter of  the  base. 

In  symbols 

L  =  \s-p, 

L  denoting  the  lateral  area,  s  the  slant 
height,  and  p  the  perimeter  of  the  base. 


Given  the  regular  pyramid 
V-ABCDE, 

Fig.  148;  the  slant  height  VK. 
To  prove  that  L  =  ^s  •  p. 


Fig.  148 


Proof:   Show  that    L  =  Js  •  AB  +  Js  •  BC  +  Js  •  CD 
+  etc. 


AREAS  OF  SURFACES  257 

271.  Theorem:  The  lateral  area  of  the  frustum  of  a 
regular  pyramid  is  equal  to  one-half  the  product  of  the  sum 
of  the  perimeters  of  the  bases  by  the  slant  height, 
or  in  symbols, 

£=i(A+A)«. 

Prove. 

EXERCISES 

1.  The  slant  height  of  a  regular  triangular 

pyramid  is  8  feet.    The  side  of  the  base  is  3  feet.    Find  the 
lateral  area. 

2.  The  altitude  of  a  regular  pyramid  is  5  feet.  The  base 
is  a  regular  hexagon  whose  side  is  6  feet.    Find  the  lateral  area. 

3.  The  altitude  of  a  regular  pyramid  is  8  feet.  The  base 
is  a  square  whose  area  is  25  square  feet.    Find  the  lateral  area. 

4.  The  base  of  a  regular  pyramid  is  a  square  whose  side  is 
6.  The  slant  height  makes  an  angle  of  45°  with  the  plane  of 
the  base.     Find  the  lateral  area. 

5.  The  base  of  a  regular  pyramid  is  a  square  whose  area  is 
900.    The  altitude  is  12.    Find  the  lateral  area. 

6.  The  sides  of  the  bases  of  a  frustum  of  a  regular  hexagonal 
pyramid  are  6  and  14  respectively.  The  slant  height  is  20. 
Find  the  lateral  area  and  total  area. 

7.  Find  the  cost  of  painting  a  church  spire  at  the  rate  of 
20  cents  per  square  yard.  The  altitude  of  the  spire  is  80  ft.  and 
a  side  of  its  hexagonal  base  is  10  feet.    ^_ ^  27rr 


Lateral  Surface 
of  Cylinder 


272.  Lateral  area  of  a  right 
cylinder    and    of  a  right  cone. 

The   lateral  area  of  a  right 

cylinder  and  of  a  right  cone  may  FIG   150 

be  found  by  rolling  the  lateral 

surface  along  a  plane.    The  lateral  surface  of  a  right 

cylinder  is  found  to  be  a  rectangle,  Fig.  150,  whose  width 


258 


THIRD-YEAR  MATHEMATICS 


is  equal  to  the  altitude  of  the  cylinder  and  whose  length  is 
equal  to  the  length  of  the  circle  forming  the  base  of  the 
cylinder. 

Hence,  L  =  2irrh, 

where  L  is  the  lateral  area,  r  the  radius  of  the  base,  and  h 
the  altitude  of  the  cylinder. 

Similarly  the  lateral 
surface  of  a  right  cone 
is  found  to  be  a  sector 
of  a  circle,  Fig.  151, 
whose  arc  equals  the 
length  of  the  circle 
forming  the  base  of  the 
cone,  and  whose  radius  is  equal  to  the  slant  height  of  the 
cone. 

Hence  L  =  ttts. 


2   TT    r 


Fig.  151 


EXERCISES 

1.  Roll  an  oblique  cylinder  along  a  plane  and  make  a  drawing 
of  the  lateral  surface. 

2.  Make  a  drawing  of  the  lateral  surface  of  an  oblique  cone. 

3.  The  extreme  length  of  a  clothes  boiler  is  24  in.,  the  width 
is  11 J  in.,  and  the  depth  12|  inches.  The  ends  of  the  boiler 
are  semicircular.  Allowance  has  to  be  made  for  locking  as 
follows:  \\  in.  on  the  width  of  the  side  piece  and  1  in.  on 
the  length;  \  in.  all  around  the  bottom  piece. 
How  much  tin  is  required  to  make  the  boiler  ? 

273.  Lateral  area  of  a  frustum  of  a  right 
cone.  Show  that  the  lateral  surface  of  a 
frustum  of  a  right  cone,  Fig.  152,  is  the 
difference  of  the  lateral  surfaces  of  two  right 
cones.  Fig.  152 


AREAS  OF  SURFACES  259 

Hence  the  lateral  area  is  given  by  the  formula 

L  =  7r(s2^2  —  $it*i). 

Since  si  and  s2  are  not  parts  of  the  frustum,  this  formula 
will  be  changed  to  a  different  form,  as  follows: 

*  =  i2     Why? 
n    r2 

.' .  S2n  —  Sir2  =  0     Why  ? 

. * .  L  =  7T  (§2^2  +  S2n  —  Sif2  —  Sifi) 

=  7r[s2  (r2 + ri)  —  si  (r2  -f  n)  ] 
=  7r(r2+ri)(s2--Si) 
.*.  I  =  ir(ri+r2)s 

This  may  be  written : 

L=4(2Trri+2irr2)s. 

Hence  the  lateral  area  of  a  frustum  of  a  right  circular 
cone  is  equal  to  one-half  the  product  of  the  slant  height  and 
the  sum  of  the  perimeters  of  the  bases. 

EXERCISES 

1.  Show  that  the  total  area  of  a  cylinder  of  revolution  is 
given  by  the  formula  T  =  2irr(h+r),  h  being  the  altitude  and  r 
the  radius  of  the  base. 

2.  Show  that  the  total  area  of  a  cone  of  revolution  is  given 
by  the  formula  T=irr(s+r),  s  being  the  slant  height  and  r  the 
radius  of  the  base.  State  in  words  the  law  expressed  by  this 
formula. 

3.  Show  that  the  lateral  area  of  a  frustum  of  a  cone  of  revolu- 
tion is  equal  to  the  slant  height  multiplied  by  the  length  of  a 


260 


THIRD-YEAR  MATHEMATICS 


circle  obtained  by  cutting  the  frustum  by  a  plane  at  equal 
distances  from  the  bases. 

Show  that  r  =  f(ri+r2),  Fig.  153. 

Hence  ri+r2  =  2r. 

Substituting  this  in  the  equation  L  =  7r(ri+r2)s, 
it  follows  that 

£=2irr.s.  Fig.  153 


4.  How  much  metal  is  required  to  construct  a  galvanized 
iron  pail  which  is  9  in.  in  diameter  at  the  top,  8j  in.  at  the 
bottom,  and  11  in.  in  the  slant  height,  allowing  lj  in.  on  the 
width,  1  in.  on  the  length  of  the  side  piece  for  locking,  and  1  in. 
on  the  diameter  of  the  bottom  piece  ? 

6.  The  lateral  area  of  a  frustum  of  a  right  circular  cone  is 
607r  square  feet.  If  the  radii  of  the  bases  are  4  ft.  and  6  ft.  re- 
spectively, find  the  slant  height. 

274.  Similar  cylinders.  Two  right  circular  cylinders 
are  similar  if  they  are  generated  by  revolving  two  similar 
rectangles  about  corresponding  sides,  Fig.  154. 


Fig.  154 


Fig.  155 


275.  Similar  cones.  Two  right  circular  cones  are 
similar  if  they  are  generated  by  revolving  two  similar 
right  triangles  about  corresponding  sides,  Fig.  155. 


AREAS  OF  SURFACES 


261 


276.  Theorem:  The  lateral  areas,  or  the  total  areas,  of 
similar  right  circular  cylinders,  or  cones,  are  'proportional 
to  the  squares  of  the  altitudes,  or  to  the  squares  of  the  radii 
of  the  bases. 

Proof:  Denoting  the  radii  by  r  and  r',  Fig.  154,  the 
altitude  by  h  and  h' ,  the  lateral  areas  by  L  and  U,  and  the 
total  areas  by  T  and  Tr,  show  that 

L  _  2-irrh  _  rh  _r     h  _  r2  _h2 
V~2Trr,h,~7h'~r,y"h,~rT2~V2' 

T       2irr(h+r)        r(h+r)   =r      h+r 
V~2>irr'(h'+r')  ~  rf  {h' +r')~  r,X  h' +r' ' 

Since  77  =  -,,  it  follows  that  tj- — ,  =  -,  =  t}  • 
h      r  h+r     r     h 

By  substitution, 

r    r'2    h'2' 

The  proof  for  the  cones,  Fig.  155,  is  similar  and  is  left 
to  the  student. 


EXERCISES 

1.  Show  that  the  lateral  areas,  or  total  areas, 
of  two  similar  right  cones  are  to  each  other  as 
the  squares  of  the  slant  heights. 

2.  How  many  square  feet  of  surface  are 
there  in  a  tank  formed  by  a  cylinder  and 
cone  of  the  dimensions  and  shape  shown  in 
Fig.  156? 


Fig.  156 


262 


THIRD-YEAR  MATHEMATICS 


3.  How  many  square  feet  of  material,  not  allowing  for 
waste,  have  been  used  in  the  construction  of  a  silo,  Fig.  157, 
the  diameter  of  whose  base  is  16  ft., 
whose  total  height  is  24  ft.,  and  the 
height  of  whose  roof  is  8  feet  ? 

GENERAL   EXERCISES 

J277.  Solve  the  following 
problems: 

1.  Find  the  lateral  surface  and 
the  total  surface  of  a  cylinder  of 
revolution  if  ft  =  8. 5  in.  and  r  =  5.3 

inches. 

2.  Find  the  lateral  surface  of  a 
quadrangular  right  pyramid,  the  side 
of  whose  base  is  7  cm.  and  whose 
altitude  is  6.8  centimeters. 

3.  Find  the  lateral  surface  and 
the  total  surface  of  a  right  cone 
whose  radius  is  4.2  and  whose  alti- 
tude is  5.7. 

4.  Find  the  lateral  surface  of  a  Fig.  157 
frustum  of  a  pyramid  whose  altitude 

is  10  and  whose  bases  are  squares  with  sides  equal  to  4  and  6 
respectively. 

5.  The  altitude  of  a  right  prism  is  40.  The  base  is  a  right 
triangle  having  the  sides  of  the  right  angle  equal  to  36  and  43 
respectively.     Find  the  lateral  and  total  area. 

6.  The  base  of  a  right  prism  is  a  regular  hexagon  whose  side  is 
6.  The  altitude  of  the  prism  is  10.   Find  the  lateral  and  total  area. 

7.  The  curved  surface  of  a  cylindrical  column  made  of 
granite  is  to  be  polished.  What  will  be  the  expense  at  the  rate 
of  60  cents  per  square  foot  if  the  diameter  of  the  base  is  4 . 5  ft. 
and  the  column  is  24  ft.  high  ? 


AREAS  OF  SURFACES 


263 


8.  The  great  pyramid  of  Cheops  is  about  460  ft.  high.  The 
base  is  a  square  whose  side  is  746  ft.  long.  What  is  the  lateral  area? 

9.  A  steeple  is  of  the  form  of  a  regular  hexagonal  pyramid. 
The  perimeter  of  the  base  is  60  feet.  The  slant  height  is  48  feet. 
How  many  square  feet  must  be  allowed  for  slating  the  steeple  ? 

10.  At  26  cents  a  square  yard  what  will  be  the  cost  of  paint- 
ing a  gas  tank  of  the  form  of  a  right  circular  cylinder  if  the  height 
is  72  ft.  and  the  diameter  of  the  base  45  feet  ? 

11.  A  funnel  is  8  in.  in  diameter  at  the 
widest  end,  lj  in.  at  the  spout,  and  1  in.  at  the 
smaller  end  of  the  spout.  The  slant  height  of 
the  funnel  is  6  in.  and  that  of  the  spout  is 
4  inches.  Allowing  for  locking  J  in.  on  the 
length  and  width  of  each  part,  find  the  amount 
of  tin  needed  to  make  the  funnel. 

12.  How  far  from  the  vertex  of  a  right 
circular  cone  must  a  plane  be  passed  parallel  to 
the  base  and  so  that  the  lateral  area  of  the  small 
cone  cut  off  shall  be  .equivalent  to  the  lateral 
area  plus  one  base  of  a  right  circular  cylinder  ? 
The  altitude  of  the  cone  is  12  in.,  the  radius  of 
the  base  8  inches.  The  altitude  of  the  cylinder 
is  4  in.  and  the  radius  of  its  base  is  2  inches. 
How  far  from  the  vertex  of  the  cone  must  the 
plane  be  passed  ? 

13.  A  windmill 
water-supply  tank, 
Fig.  158,  is  8  ft.  in 
diameter  and  12  ft. 
high.  The  roof  is 
9  ft.  in  diameter 
and  3  ft.  high. 
How  much  mate-  Fig.  159 
rial  was  used  in  its  construction  ? 

14.  The  width  and  length  of  a  tent,  Fig.  159,  are  12  ft.  and 
18  ft.  respectively.    The  height  of  the  pole  is  8  ft.  and  the  height 


Fig.  158 


264 


THIRD-YEAR  MATHEMATICS 


of  the  wall  3|  feet.    How  much  material  was  used  in  making  the 
tent? 

Surfaces  of  Revolution 

278.  Surface  of  revolution.  If  a  line  segment,  AB, 
Figs.  160-164,  revolves  about  a  straight  line,  CD,  in  the 
same  plane  as  an  axis,  every  point  of  the  segment  describes 
a  circle  whose  plane  is  perpendicular  to  the  axis.    Why  ? 

The  surface  generated  by  the  segment  is  a  surface  of 
revolution.  According  to  the  position  of  the  segment  with 
reference  to  the  axis,  the  surface  of  revolution  of  the 
segment  is  a  lateral  surface  of  a  right  cone,  Fig.  160,  a 


Fig.  163 


frustum  of  a  right  cone,  Fig.  161,  a  right  cylinder,  Fig.  162, 
a  surface  of  a  circle,  Fig.  163,  or  a  circular  ring, 
Fig.  164. 


AREAS  OF  SURFACES 


265 


279.  Theorem:  If  half  of  a  regular  polygon  having 
an  even  number  of  sides  is  revolved  about  a  diagonal 
joining  two  opposite  vertices,  the  area  of  the  surface 
thus  generated  is  equal  to  the 
product  of  the  diagonal  by  the 
length  of  the  circle  inscribed  in 
the  polygon.  C} 

Proof:  The  surface,  Fig.  165, 
is  composed  of  cones,  frus- 
tum of  cones,  and  cylinders. 
Hence  the  area  may  be 
found  by  adding  the  areas 
of  these  cones,  frustums,  and 
cylinders. 

It  will  be  shown  that 
one  formula  may  be  used  to 
find  the  lateral  surface  of 
each. 

1.  The  area  of  the  surface 
generated  by  AB,  Fig.  166,  is 
given  by 


Fig.  165     • 


L^wBB  XAB,  §272. 

Bisect  AB  at  M. 

Draw  MM'±AF. 

Show  that    MM'  =  \BB'. 


Then  Ll  =  2irMM'XAB. 

Draw  MO±AB. 

AOMM'oAABB'. 
.   AB     AB' 
"MO    MM'' 


.\MM'XAB  =  MOxAB'. 

/.L1  =  2ttMOxAB7. 


F 
Fig.  166 


Why? 

Why? 
Why? 

Why? 
Why? 


266 


THIRD-YEAR  MATHEMATICS 


2.  The  area  of  the  surface  generated  by  BC,  Fig.  167, 
is  given  by 

L2  =  7r(CC'+££')£C,  §273.      . 


Bisect  BC  at  M .     Draw  MM'±AF. 
Show  that  CC'+BB'  =  2MM'. 
Then 


Draw 
Then 


U  =  2ttMM,XBC. 

BB"±CC. 

ACBB"i*AOMM\ 

Why? 

CB     BB"      B'C 

*  *  MO    MM'    MM' ' 


MM'XCB  =  M0XB'C. 


:.U  =  2irM0xB,C    c 

M 

3.  The  area  of  the  surface  generated    D 
by  CD,  Fig.  168,  is  given  by 


Lz  =  2irDD'XCD. 

Bisect  CD  at  M.    Draw  MO. 

Then  Ls  =  2tM0XCD. 

Similarly  the  area  of  the  remaining  part  of  the  surface 


is  found. 
Thus, 


Adding, 
Fig.  165, 


Ll  =  2wM0xAB' 
U  =  2TvM0XBrC' 


U  =  2ivM0xC,D,i  etc. 

L  =  2TMd(AB'+B'C'+. .  ..E'F), 

or    L  =  2ttMOXAF 


AREAS  OF  SURFACES 


267 


280.  Area  of  the  surface  of  a  sphere.  To  find  the  area 
of  the  surface  of  a  sphere  inscribe  in  a  semicircle  half  a 
regular  polygon  as  ABODE,  Fig.  169. 

The  area  of  the  surface  generated  by  the  polygon 
ABCDE  is  2tMOxAE. 

Let  the  number  of  sides  of  the  polygon 
be  increased  indefinitely. 

Then  the  polygon  approaches  the  circle 
as  a  limit. 

The  area  of  the  surface  generated  by 
the  polygon  approaches  as  a  limit  the  area 
of  the  surface  of  the  sphere  generated  by 
the  semicircle. 

MO  .approaches  the  radius  r  as  a  limit. 

Hence  2wMO  approaches  2wr,  and 
approaches  2irrXAE,  which  is  equal  to  27rrX2r  =  47rr2. 
Hence  the  area  of  the  surface  generated  by  ABCDE 
approaches  47rr2  as  a  limit. 

Thus  the  preceding  discussion  leads  to  the  following 
theorem: 

The  area  of  the  surface  of  a  sphere  is  equal  to  the  product 
of  the  diameter  by  the  length  of  a  great  circle,  or 


Fig.  169 


2ttMOXAE 


S  =  4irr2. 

281.  Zone.  A  portion  of  a 
spherical  surface  included  between 
two  parallel  planes  is  a  zone, 
Fig.  170.  The  distance  between 
the  planes  is  the  altitude  of  the 
zone.  The  sections  made  by  the 
planes  are  the  bases  of  the  zone.  If 
one  of  the  planes  is  tangent  to  the 
spherical  surface,   the  zone  is  said  to  have  one  base. 


Fig.  170 


268  THIRD-YEAR  MATHEMATICS 

EXERCISES 

1.  Show  that  the  area  of  a  zone  is  equal  to  the  product  of  the 
altitude  by  the  length  of  a  great  circle,  or  Z  =  2-n-rh.  Determine  the 
area  of  a  zone  whose  altitude  is  12  if  the  radius  of  the  sphere 
is  15. 

2.  The  areas  of  two  spherical  surfaces  are  to  each  other  as  the 
squares  of  the  radii.     Prove. 

3.  How  many  square  feet  should  be  allowed  for  polishing  a 
hemispherical  dome  whose  diameter  is  if  feet  ? 

4.  The  earth  is  approximately  a  sphere  of  diameter  equal 
to  7,920  miles.    How  large  is  its  surface  ? 

6.  Find  the  area  of  the  north  temperate  zone,  assuming  its 
altitude  to  be  about  1,800  miles. 

6.  Show  that  the  surface  of  a  sphere  is  equal  to  the  lateral 
surface  of  the  circumscribed  cylinder. 

7.  Find  the  ratio  of  the  area  of  the  surface  of  the  moon  to  that 
of  the  earth,  assuming  the  diameter  of  the  moon  to  be  2, 162  miles. 

8.  Two  parallel  planes,  equidistant  from  the  center  of  a 
sphere  of  radius  r,  cut  from  the  sphere  a  zone  whose  area  is  \  the 
area  of  the  curved  surface  of  the  cylinder  having  the  same  bases 
as  the  zone.  Find  the  distance  of  the  planes  from  the  center  of 
the  sphere.     (Harvard.) 

9.  How  far  in  one  direction  can  a  man  see  from  the  top  of 
Mount  Etna  ? 

The  required  distance  is  the  geometric  mean  between  the  height 
of  the  mountain,  3,300  m.,  and  the  sum  of  the  height  and  the 
diameter  of  the  earth,  6,374  kilometers. 

10.  Show  that  if  a  man  ascended  in  a  balloon  to  a  height 
equal  to  the  earth's  radius  he  would  see  one-quarter  of  the 
earth's  surface.     (Harvard.) 

11.  The  lateral  area  of  a  cone  of  revolution  and  the  area  of  a 
sphere  are  each  equal  to  49  square  feet.  If  the  radius  of  the 
sphere  equals  the  radius  of  the  base  of  the  cone,  find  the  altitude 
of  the  cone. 


AREAS  OF  SURFACES 


269 


12.  The  eight  vertices  of  a  cube  all  lie  on  a  sphere. 
Prove  that  every  diagonal  of  the  cube  is  a  diameter  of  the 
sphere.  If  one  edge  of  the  cube  is  a,  find  the  area  of  the 
zone  of  one  base  cut  off  by  the  plane  of  one  face  of  the  cube. 
(Harvard.) 

13.  If  the  temperate  zones  were  between  the  30°  and  60° 
parallels  of  latitude,  what  proportion  of  the  earth's  surface 
would  they  comprise?  Give  the  details  of  the  computation. 
(Board.) 

14.  Two  parallel  planes  on  the  same  side  of  the  center  of  a 
sphere  of  radius  r  bound  a  zone.  The  area  of  this  zone  is  one- 
fourth  that  of  the  sphere.  The  area  of  the  circle  cut  by  the  plane 
nearer  to  the  center  is  double  that  cut  by  the  farther.  Find  the 
distance  from  the  center  of  the  sphere  to  the  nearer  plane. 
(Harvard.) 

282.  The  chapter  has  taught  the  meaning  of  the  fol- 
lowing terms: 


polyedron 

face,  edge,  vertex,  surface 
of  a  polyedron 

tetraedron,  hexaedron 

octaedron,  dodecaedron, 
icosaedron 

pyramidal  and  conical  sur- 
face 

directrix,  generatrix 

triangular,  quadrangular, 
pentagonal  pyramid 

regular  pyramid 

slant  height 

circular,  right  circular  cone 

cone  of  revolution 

cylindrical  surface,  cylinder 

right  cylinder,  oblique 
cylinder 


cylinder  of  revolution 
prismatic  surface,  prism 
right  and  oblique  prism 
triangular,    quadrangular, 

etc.,  prism 
section,  right  section 
parallelopiped 
truncated  prism 
frustum  of  a  pyramid 
frustum  of  a  cone 
sections  of  a  right  circular 

cone 
circle,    ellipse,    parabola, 

hyperbola 
similar    cylinders,   similar 

cones 
surface  of  revolution 
zone 


270   x  THIRD-YEAR  MATHEMATICS 

Summary 

283.  The  truth  of  the  following  theorems  has  been 
established : 

1.  The  equation  J-\-v  =  e+2  expresses  the  relation 
between  the  number  of  faces,  vertices,  and  edges  of  a 
convex  polyedron. 

2.  The  lateral  edges  of  a  regular  pyramid  are  equal. 

3.  The  lateral  faces  of  a  regular  pyramid  are  congruent 
isosceles  triangles. 

4.  The  lateral  edges  of  a  prism  are  equal. 

5.  The  lateral  faces  of  a  prism  are  parallelograms. 

6.  The  sections  of  a  prism  made  by  parallel  planes  are 
congruent. 

7.  The  right  sections  of  a  prism  are  congruent. 

8.  A  section  of  a  prism  parallel  to  the  base  is  congruent 
to  the  base. 

9.  The  section  of  a  cylinder  made  by  a  plane  passing 
through  an  element  is  a  parallelogram. 

10.  The  sections  of  a  cylinder  made  by  parallel  planes 
cutting  all  elements  are  congruent. 

11.  The  sections  of  a  cylinder  parallel  to  the  bases  are 
congruent  to  the  base. 

12.  A  section  of  a  cone  made  by  a  plane  passing  through 
the  vertex  is  a  triangle. 

13.  A  section  of  a  circular  cone  made  by  a  plane  parallel 
to  the  base  is  a  circle. 

14.  If  a  pyramid  is  cut  by  a  plane  parallel  to  the  base, 
the  edges  and  altitude  are  divided  proportionally;  the  sec- 
tion is  a  polygon  similar  to  the  base;  the  areas  of  the  section 
and  the  base  are  proportional  to  the  squares  of  the  distances 
from  the  vertex. 


AREAS  OF  SURFACES  271 

15.  The  lateral  areas,  or  the  total  areas  of  similar  circular 
cylinders,  or  cones,  are  proportional  to  the  squares  of  the 
altitudes,  or  to  the  squares  of  the  radii  of  the  bases. 

16.  The  plane  sections  of  a  right  circular  cone  are 
two  intersecting  straight  lines,  a  circle,  a  parabola,  an 
ellipse,  and  a  hyperbola. 

284.  The  following  is  a  summary  of  the  formulas  of 
this  chapter: 

I.  Lateral  area: 

1.  Of  a  prism, 

L=pXe 

2.  Of  a  right  prism, 

L=p-h 

3.  Of  a  regular  pyramid, 

L  =  \sXp 

4.  Of  a  frustum  of  a  pyramid, 

L  =  \{p,+p,)s 

5.  Of  a  right  cylinder, 

L  =  2-rrrh 

6.  Of  a  right  cone, 

L  =  irr  •  5 

7.  Of  a  frustum  of  a  right  circular  cone, 

L=Tr(ri+r2)s 

II.  The  area  of  a  surface  generated  by  revolving  half 
of  a  regular  polygon  about  a  diagonal  joining  two  directly 
opposite  vertices, 

L  =  2ttMOxAF 

III.  The  area  of  the  surface  of  a  sphere, 

S=4-rrr2 

IV.  The  area  of  a  zone, 

Z=2irrh 


CHAPTER  XIII 


VOLUMES 
Volume  of  a  Rectangular  Parallelopiped 

285.  Volume.  To  measure  the  space  bounded  by  the 
surface  of  a  solid,  a  cube  is  used  whose  edges  are  the  unit 
of  length.  This  cube  is  said  to  be  the  unit  of  volume,  and 
the  number  of  times  it  is  contained  in  the  solid  is  the 
volume  of. the  solid. 

286.  Volume   of  a  rectangular  parallelopiped.    Let 

a,  b,  and  c  be  the  lengths  of  three  concurrent  edges, 

Fig.  171.   By  drawing  planes 

parallel   to   the   base,   the 

parallelopiped,  p,  may  be 

divided  into  a  equal  layers, 

(Z),  Fig.  172.    Each  layer,  Z, 


////// 

/ 

x 

/  /  /  /  / 

y 

'  /  /  /   /   / 

X 

rfi^V 

y 

i 

i 

c 

y 

////// 

> 

////// 

s 

'///// 

s 

/ 

r 

Fig.  171 


Fig.  172 


may  be  divided  into  b  equal  strips,  (s),  and  each  strip,  s, 
into  c  equal  cubes,  (u). 

272 


VOLUMES  273 

Hence  the  volume  of  a  strip  s  is  c,  the  volume  of  a 
layer  I  is  bXc,  and  the  volume  of  the  parallelopiped 
p  is  aXbXc. 

So  far  we  have  assumed  that  a,  6,  and  c  are  com- 
mensurable. Let  us  suppose  a,  b,  and  c  to  be  incommen- 
surable,   e.g.,    a  =  V6m;    b  =  VlOm;   c  =  Vlbm.     Then 

a=2.4494.  .  .  .  m,6=3.1623  ....  m,c=3.8730 m. 

In  this  case  the  volume  of  the  parallelopiped  may  be  deter- 
mined to  any  desired  degree  of  accuracy,  as  follows : 

1.  Taking  a  =  2. 4,  6  =  3.1,  and  c  =  3.8  and  using  1 
decimeter  as  the  unit  of  length,  we  have  7  =  24X31X38 
cubic  decimeters  =  28,272  cubic  decimeters  =  28 .  272  cubic 
meters. 

2.  Similarly,  for  a  =  2 .  44,  b  =  3 .  16,  and  c  =  3 .  87,  using 
a  centimeter  as  unit,  7  =  244X316X387  cubic  centi- 
meters =29,993,456  cubic  centimeters  =  29. 839248  cubic 
meters. 

3.  For  a  =  2.449,  6  =  3.162,  and  c  =  3.873,  using  a 
millimeter  as  unit  of  length,  7  =  29,991,497,274  cubic 
millimeters  =  29. 99 14  ....  cubic  meters. 

4.  For  a  =  2.4494,  6  =  3.1623,  and  c  =  3.8730  we 
find  7  =  29,999,241,802,260  cubic  one-tenth  millimeters 
=  29.9992  ....  cubic  meters. 

By  taking  a,  6,  and  c  to  a  still  greater  number  of 
decimal  places,  we  may  obtain  an  approximate  value  of 
7  differing  from  the  actual  value  by  a  number  less  than 
any  assigned  quantity. 

Assuming  the  formula  V  =  a  •  6  •  c  to  hold  for  incom- 
mensurable values  of  a,  6,  and  c,  we  find 

F  =  V/6Xl/10XV/15  cubic  meters 
=  ^6X10X15  cubic  meters  =  30  cubic  meters. 


274  THIRD-YEAR  MATHEMATICS 

However,  this  is  the  value  approached  by  the  sequence 
7  =  28.272,  29.8392,  29.9914,  29.9992,  etc. 

Thus,  whether  a,  b,  and  c  have  commensurable  or 
incommensurable  values,  the  preceding  discussion  shows 
that  the  volume  of  a  rectangular  parallelopiped  is  equal  to 
the  product  of  the  three  dimensions. 

EXERCISES 

Prove  the  following: 

1.  The  volume  of  a  rectangular  parallelopiped  is  equal  to  the 
product  of  the  base  by  the  altitude. 

2.  The  volume  of  a  cube  is  equal  to  the  cube  of  an  edge. 

3.  The  volumes  of  two  cubes  are  to  each  other  as  the  cubes  of  the 
edges. 

4.  Two  rectangular  parallelopipeds  are  to  each  other  as  the 
products  of  the  three  dimensions. 

5.  Two  rectangular  parallelopipeds  having  equal  altitudes 
(bases)  are  to  each  other  as  the  bases  (altitudes). 

6.  Two  rectangular  parallelopipeds  having  two  (one)  dimen- 
sions equal  are  to  each  other  as  the  third  (product  of  the  other  two) 
dimension. 

7.  Show  that  the  volume  of  a  cube  varies  directly  as  the 
cube  of  the  edge.  If  the  edge  of  a  given  cube  is  doubled,  trebled, 
etc.,  how  does  the  volume  of  the  new  cube  compare  with  that  of 
the  given  cube  ? 

8.  How  many  dimensions  are  needed  to  determine  the 
volume  of  a  rectangular  parallelopiped  ?    The  area  of  one  face  ? 

9.  A  room  is  12 . 5  ft.  long,  12  ft.  wide,  and  11  ft.  high.  Find 
how  many  cubic  feet  of  air  it  contains. 

10.  How  many  bricks  will  be  needed  to  build  a  wall 
20X4X2  ft.,  making  no  allowance  for  mortar? 

Assume  the  size  of  a  brick  to  be  9  X3  X4§  inches. 


VOLUMES 


275 


Comparison  of  Volumes 

287.  Theorem:  The  plane  passed  through  two  diago- 
nally opposite  edges  of  a  right  parallelopiped  divides  the 
parallelopiped  into  two  equal  triangular  right  prisms. 

Given  the  right  parallelopiped 
AG,  Fig.  173,  plane  ACGE  passed 
through  AE  and  CG. 

To  prove  that  prism 

ABC-F  §2  prism  CDA-H. 

Proof:   AABC&ACDA.     Why? 

Imagine  ABC-F  placed  on  CDA-H  making  AABC 
coincide  with  AC  DA. 

Then  BF  will  coincide  with  DH,  AE  with  GC,  and  GC 
with  AE.    Why  ? 

Hence  AEFG  will  coincide  with  AGHE.     Why  ? 

.*.  Prisms  ABC-F  and  CDA-H  coincide  throughout 
and  are  congruent. 

288.  Theorem:  An  oblique  prism  is  equal  to  a  right 
prism  whose  base  is  equal  to  a  right  section  of  the  oblique 
prism,  and  whose  altitude  is  equal  to  a  lateral  edge  of  the 
oblique  prism. 

Given  the  oblique  prism  AD',  Fig.  174,  and  the  right 
prism  FT,  FI  being  a  right  section  of 
prism  AD';   FF'  =  AA'. 

To  prove  that  AD'  =  Fr. 

Proof:  Imagine  the  truncated  prism 
AI  to  be  placed  on  the  truncated 
prism  AT  making  AD  coincide  with 
A'D'. 

Show  that  FA,  GB,  HC,  etc.,  coin- 
cide respectively  with  F' A' ,  G'B', 
H'C,  etc. 


py 


276  THIRD-YEAR  MATHEMATICS 

Show  that  AG,  BH,  etc.,  coincide  respectively  with 
AW,  B'H',  etc. 

Hence     AI  =  A'Ir,  since  they  can  be  made  to  coincide. 

But         AI'mAI' 

.'.    AD'  =  Ff 

(Equals  subtracted  from  equals  give  equals.) 

Exercises 

1.  The  diagonal  of  a  cube  is  l$Vz.    Find  the  volume. 

2.  Find  the  surface  and  volume  of  a  cube  whose  diagonal 
is  24  inches. 

3.  How  many  gallons  of  water  are  contained  in  a  tank  whose 
shape  is  that  of  a  rectangular  parallelopiped  whose  dimensions 
are  12,  20,  and  10.4  feet  respectively? 

A  gallon  contains  231  cubic  inches. 

4.  Given  a  sphere  whose  diameter  is  10  inches.  Find  the 
volume  and  the  surface  of  the  inscribed  cube.     (Sheffield.) 

Volume  of  a  Prism 

289.  Theorem :  The  volume  of  a  right  triangular  prism  is 
equal  to  the  product  of  the  base  by  the  altitude. 


Given  the  right  triangular  prism 
ABC-F,  Fig.  175. 

j^\ 

To  prove  that 

ABC-F=ABCXCF. 

A 

Proof:    Draw    FG±DE  and 

H 

Fig.  175 

CH±AB. 

Show  that  FG  and  CH  are  both  perpendicular  to  plane 
AE. 

Pass  a  plane  through  FG  and  CH. 

Draw  AK  \\  EC,  CK  \\  HA.  Draw  KI  \\  HG,  meeting 
plane  DGF  in  /. 


VOLUMES  277 

Then  AHCK-F  is  a  rectangular  parallelopiped. 

AHC-F  =  iAHCK-F    (§287). 
.-.    AHC-F  =  \AHCKXCF    (§286). 
.*.     AHC-F  =  AHCXCF.     Why? 

Similarly,  prove  that 

hbc-f=hbcxcf: 

Adding,  ABC-F=ABCXCF. 

290.  Theorem:    The  volume  of  a  right  parallelopiped 
is  equal  to  the  product  of  the  base  by  the  altitude. 

Divide  the  parallelopiped  into  two  equal  right  triangular 
prisms  (§  287).     Then  find  the  sum  of  the  two  triangular  prisms. 

291.  Theorem:     The  volume  of  an  oblique  parallel- 
opiped is  equal  to  the  product  of  the  base  by  the  altitude. 

Given  the  oblique  parallelopiped  AG,  Fig.  176,  whose 

base  is  ABCD  and 

h        h' g g' 


whose  altitude  is  h. 
To   prove  that 
AG=ABCDXh. 


Proof:    Con-      A  ^  b  ~b' 

struct   the  right  yig.  176 

section  A' H' '. 

On  A'H'  as  base  construct  the  right  parallelopiped 
A'G',  having  its  edge  A'B'  equal  to  AB. 
Then  AG  =  A'G'.     Why? 

But  A'G'  =  A'H'XA'B' 

=  (hXA'D')XA'B' 
=  hXA'D'XAB 
=  hX(A'D'XAB) 
=  hXABCD. 
Briefly,  this  result  may  be  expressed  by  the  equation 

V=h>b. 


278 


THIRD-YEAR  MATHEMATICS 


292.  Theorem:  The  plane  passed  through  two  diago- 
nally opposite  edges  of  any  parallelopiped  divides  the  parallel- 
opiped into  two  equal  triangular  prisms. 

Draw  the  right  section 
IJKL,  Fig.   177. 

Show  that  the  triangular 
prism  ABC-F  =  the  right  prism 
having  the  base  UK  and  the 
altitude  equal  to  BF. 

Show  that  CDA-H  =  the 
right  prism  having  the  base 
KLI  and  the   altitude  equal  to  BF. 

Show  that  these  two  right  prisms  are  equal  to  each 
other  (§  289). 

.-.    ABC-F  =  CDA-H. 


rr               s 

* 

IF 

Dl 

'^9* 

1  / 

V 

Fig.  177 


293.  Theorem:  The  volume  of 
any  triangular  prism  is  equal  to  the 
product  of  the  base  by  the  altitude. 

Construct  the  parallelogram  pIG   -^g 

ABCD,  Fig.  178. 

Construct  the  parallelopiped  ABDC-B'. 
Then  ABC-B'  =  ^ABDC-B'     (§  292). 

ABDC-B' =  ABDCXh     (§291). 
ABC-B' =  %ABDCXh 
=  ABCXh. 

294.  Theorem:  The  volume  of  any 
prism  is  equal  to  the  product  of  the 
base  by  the  altitude. 

By  drawing  planes  through 
AA',  Fig.  179,  and  all  the  other 
lateral  edges,  the  prism  may  be 
divided  into  triangular  prisms, 
having  the  same  altitude,  h,  and   bases  6i,  62,  b3,  etc 


VOLUMES  279 

Then  ABC-A'  =  hh, 

ACD-A'  =  b2h, 
APE  -  A'  =  bzh,  etc. 

Adding,    ABODE- A' =  (6i+62+63+  etc.)h. 
This  result  may  be  expressed  briefly  by  means  of  the 
equation  y=b-h 

EXERCISES 

1.  Prisms  having  equal  bases  and  altitudes  are  equal.    Prove. 

2.  The  volumes  of  two  prisms  having  equal  bases  are  to  each 
other  as  the  altitudes.     Prove. 

3.  How  many  cubic  yards  of  earth  must  be  removed  to  build 
a  trench  200  ft.  long  and  10  ft.  deep,  4  ft.  wide  at  the  bottom  and 
6  ft.  at  the  top  ? 

4.  What  will  be  the  cost,  at  42  cents  a  cubic  yard,  to  dig  a 
ditch  10  rd.  long,  4  ft.  deep,  7  ft.  wide  at  the  top,  and  4^  ft.  wide 
at  the  bottom  ? 

5.  Find  the  volume  of  a  right  triangular  prism,  8  in.  high, 
whose  base  is  an  equilateral  triangle  with  sides  of  2  inches. 

6.  A  triangular  prism  is  10  ft.  high.  The  base  is  a  right 
triangle  whose  sides  are  3,  4,  and  5  ft.  respectively.  Find  the 
volume. 

7.  The  volume  of  a  triangular  prism  is  250.  The  base  is  an 
equilateral  triangle  whose  side  is  7.  Find  the  altitude  of  the 
prism. 

8.  Find  the  volume  of  a  triangular  prism  whose  height  is 
30  in.  and  the  sides  of  whose  base  are  12  in.,  10  in.,  and  10  inches. 

9.  Find  the  volume  of  a  prism  whose  base  is  a  rhombus,  one 
of  whose  sides  is  40  in.,  and  whose  shorter  diagonal  is  48  inches. 
The  height  of  the  prism  is  60  inches. 

10.  A  regular  hexagonal  prism  has  the  area  of  one  base  12  and 
its  total  area  276.     Find  the  volume  of  the  prism.     (Yale.) 


280 


THIRD-YEAR  MATHEMATICS 


295.  Inscribed  prism,  pyramid,  and  frustum  of  a 
pyramid.  A  prism,  a  pyramid,  and  a  frustum  of  a  pyra- 
mid are  said  to  be  inscribed  if  the  lateral  edges  are  ele- 
ments of  a  cylinder,  a  cone,  or  a  frustum  of  a  cone, 
respectively,  and  if  the  bases  of  the  former  are  inscribed 
in  the  bases  of  the  latter,  Fig.  180.     It  will  be  assumed  that 


Fig.  180 


the  volume  of  an  inscribed  prism,  pyramid,  or  frustum 
of  a  pyramid  is  less  than  the  volume  of  the  cylinder,  cone, 
and  frustum  of  a  cone  respectively. 

296.  Tangent  plane.  If  a  plane 
contains  one,  and  only  one,  element  of  a 
cylinder,  a  cone,  or  a  frustum  of  a  cone, 
but  does  not  intersect  the  surface,  it  is 
a  tangent  plane,  Fig.  181. 

EXERCISES 

1.  Prove  that  a  plane  passing  through  a  tangent  to  the  base 
of  a  circular  cone  and  the  element  drawn  through  the  point  of 
contact  is  tangent  to  the  cone. 

Is  this  theorem  necessarily  true  when  the  cone  is  not  circular  ? 
(Harvard.) 

2.  The  intersection  of  two  planes  tangent  to  a  circular 
cylinder  is  parallel  to  the  elements  of  the  cylinder.     (Yale.) 


VOLUMES 


281 


297.  Circumscribed  prism,  pyramid,  and  frustum  of  a 
pyramid.  A  prism,  a  pyramid,  and  a  frustum  of  a  pyra- 
mid are  said  to  be  circumscribed  if  the  lateral  faces  are 
tangent  to  the  lateral  surface  of  a  cylinder,  a  cone,  and  a 
frustum  of  a  cone,  respectively,  and  if  the  bases  of  the 
former  are  circumscribed  about  the  bases  of  the  latter, 
Fig.  182.     It  will  be  assumed  that  the  volume  of  a  circum- 


/C  yi 

I ' ' 

iii  j 

iJ 

IV 

r 

Fig.  182 

scribed  prism,  pyramid,  or  frustum  of  a  pyramid  is  greater 
than  the  volume  of  the  cylinder,  cone,  or  frustum  of  a 
cone  respectively. 

Volume  of  a  Cylinder 

298.  Theorem:    The  volume  of  a  circular  cylinder  is 
equal  to  the  product  of  the  base  by  the  altitude. 

Given  the  cylinder  AC,  Fig.  183, 
whose  altitude  is  h  and  whose  base 
is  b. 

To  prove  that  the  volume,  v  =  b-h. 

Proof  (indirect  method) : 

Assume  that  v^b'h.  Then  v > bh , 
or  v<bh. 

First,  assume  v <bh,  or  v  =  Bh,  where  b> B. 

Inscribe  in  the  cylinder  a  prism  whose  base,  B' ,  is 
greater  than  B,  i.e.,  such  that  b>B'>B. 


Fig.  183 


282  THIRD- YEAR  MATHEMATICS 

Then  B,h>Bh. 

Thus  B'h,  the  volume  of  the  inscribed  prism,  is  greater 
than  Bh,  the  volume  of  the  cylinder. 

This  is  impossible,  and  v  is  not  less  ^ ^\ 

than  b'h.  .  A.  _^# 

Secondly,  assume  v>bh,  or  v  =  Bh,        J!  II 

where  b<B.  IBh  I 

Circumscribe  about  the  cylinder  a  llrV — ^J 
prism  whose  base,  B',  is  less  than  B,  A%j^b^jU 
i.e.,  such  that  b<B'<B.  FlG   183 

Then  B'h<B-h. 

Thus  the  volume  of  the  circumscribed  prism  is  less  than 
the  volume  of  the  cylinder  which  was  assumed  to  be  equal 
to  B-h.     ' 

This  is  impossible,  and  v  is  not  greater  than  b*h. 

/.    v  =  b-h. 

299.  Theorem:  The  volume  of  a  circular  cylinder  of 
revolution  is  given  by  the  formula 

where  h  is  the  altitude  and  r  the  radius  of  the  base.     Prove. 

EXERCISES 

1.  How  many  cubic  yards  of  dirt  must  be  removed  in  the 
excavation  of  a  tunnel  20  ft.  high  and  f  mi.  long  ?  The  shape 
of  the  tunnel  is  to  be  such  as  to  make  the  cross-section  a  semi- 
circle. 

2.  A  cubic  foot  of  copper  is  drawn  into  a  wire  J  in.  in  diam- 
eter.   What  is  the  length  of  the  wire  ? 

3.  The  inner  diameter  of  a  pipe  75  yd.  long  is  2f  inches. 
How  many  feet  of  water  does  it  contain  ? 

4.  The  cost  of  digging  a  well  is  $3 .  25  per  cubic  yard.  What 
will  it  cost  to  dig  a  well  80  ft.  deep  and  5  ft.  in  diameter  ? 


VOLUMES  283 

5.  Find  the  volume  of  a  circular  ring  whose  inner  radius  is 
8  in.  and  whose  outer  radius  is  10  inches. 

Consider  the  ring  as  a  cylinder  whose  height  is  the  length  of  a 
circle  whose  radius  is  the  mean  of  the  inner  and  outer  radii. 

6.  What  must  be  the  height  of  a  water  boiler  holding  30  gal. 
if  its  diameter  is  1  foot  ?  i 

7.  The  altitude  of  a  cylinder  of  revolution  is  26.  The  radius 
of  the  base  is  24.     Find  the  volume. 

8.  The  volumes  of  two  similar  cylinders  of  revolution  are  to 
each  other  as  the  cubes  of  the  altitudes,  or  as  the  cubes  of  the  radii  of 
the  bases.    Prove. 

V      itr^h      r^      h      r^     r     r^ 
Show  that -----Xj^Xp-pi . 


9.  A  cylindrical  tank,  Fig.  184,  is  partly  /X-^.^X 
filled  with  water.    The  tank  is  4  ft.  long  and  (  $j  "T    } 

10  in.  in  diameter.     If  the  greatest  depth  of  \!Z \~S 

the  water  is  8  in.,  how  much  of  the  tank  is  Fig.  184 

filled  with  water  ? 

10.  A  rectangle  is  rotated  about  one  of  its  sides  as  an  axis. 
What  is  the  ratio  of  the  volumes  generated  by  the  triangles  into 
which  the  rectangle  is  divided  by  one  of  its  diagonals  ?    (Board.) 

Use  §  306. 

11.  A  cube,  5  in.  on  a  side,  is  just  covered  by  water  in  the 
bottom  of  a  cylindrical  pail  12  in.  in  diameter.  How  high  will 
the  water  stand  in  the  pail  when  the  cube  is  removed? 
(Harvard.) 

12.  Compare  the  volumes  of  cylinders  of  altitude  8  in.  and 
10  in.,  respectively,  whose  convex  surface  can  be  exactly  covered 
by  a  rectangular  sheet  of  paper  8  by  10  in.  in  size.     (Yale.) 

13.  Prove  that  if  a  square  be  rotated  completely  around  a 
straight  fine  in  its  plane,  not  crossing  it,  but  parallel  to  one  of 
its  sides,  the  volume  generated  is  equal  to  the  product  of  the 
area  of  the  square  and  the  length  of  the  circumference  traced 
by  its  center.     (Harvard.) 


284 


THIRD-YEAR  MATHEMATICS 


14.  A  glass  vessel  made  in  the  form  of  a  right  circular  cylinder 
contains  a  certain  amount  of  water.  The  diameter  of  the  base 
of  the  vessel  is  5  inches.  When  an  irregular  mass  of  gold  is 
dropped  into  the  vessel  it  is  entirely  covered  by  water  and  the 
level  of  the  water  rises  3  inches.  What  is  the  weight  in  ounces  of 
the  lump  of  gold  if  gold  weighs  11  oz.  per  cubic  inch  ?     (Board.) 

15.  A  regular  hexagonal  prism  is  inscribed  in  a  right  circular 
cylinder  whose  base  has  radius  10.  Compare  their  lateral  areas 
and  their  volumes.     (Yale.) 


Volume  of  a  Pyramid 

300.  Theorem:  If  two  pyramids  have  equal  bases  and 
equal  altitudes,  sections  made  by  planes  parallel  to  the  bases 
and  at  equal  distances  from  the  vertices  are  equal. 


Fig.  185 


Proof: 


A'B'C    OE 


>2 


(§  260) 


ABC       CW 

M'N'P'Q'R'  =  W'2 
MNPQR       sf2 

OE,2  =  ST'2 
OE2     OT2 
A'B'Ci^M'N'P'Q'R' 
•*      ABC  =     MNPQR 
.\    A'B'C  =  M'N'P'Q'R' 


Why? 

Why? 
Why? 


VOLUMES 


285 


301.  Theorem:  If  two  triangular  pyramids  have  equal 
bases  and  altitudes,  their  volumes  are  equal. 


Fig.  186 


Discussion:  Place  the  bases  of  the  pyramids,  Fig.  186, 
in  the  same  plane  and  divide  the  altitude  h  into  equal 
parts,  x. 

Through  the  points  of  division  draw  planes  parallel  to 
the  plane  of  the  bases,  cutting  the  pyramids  into  sections 
equal  in  pairs  (§  300). 

Using  each  section  as  lower  base,  construct  prisms 
whose  altitudes  are  equal  to  x  and  whose  lateral  edges  are 
parallel  to  AV  and  A'V  respectively.  In  this  way  the 
prisms  P,  Q,  R,  and  S  have  been  constructed  in  pyramid 
V-ABC.  Imagine  a  similar  set  of  prisms  constructed 
in  V'-A'B'C. 

Using  each  section  as  upper  base,  construct  prisms 
whose  altitudes  are  equal  to  x  and  whose  lateral  edges  are 
parallel  to  AV  and  A'V  respectively.  This  gives  prisms 
P',  Q',  and  R'  in  the  pyramid  V'-A'B'C  Imagine  a 
similar  set  of  prisms  constructed  in  V-ABC. 

Show  that  P  =  P',  Q  =  Q',  R  =  R'. 

Denote  P+Q+R+S  by  X  and  P'+Q'+R'  by  F. 

ThenZ-F  =  >S. 


286  THIRD-YEAR  MATHEMATICS 

Denoting  the  volumes  of  V-ABC  and  V'-A'B'C  by 
V  and  V,  respectively,  we  have 

X>V>Y 

and  X>V>Y. 

.'.  the  difference  between  V  and  V  is  less  than  the 
difference  between  X  and  Y, 

i.e.,  V-V'<X-Y, 

or  V-V'<S. 

By  dividing  the  altitude  h  into  twice  as  many  parts  as 
before,  a  new  set  of  prisms  is  obtained  for  which  the 
same  relations  hold  and  V—  V'<Sif  where  Si  is  half  as 
large  as  S.  This  process  can  be  repeated  as  often  as 
required,  and  an  Sn  obtained  as  small  as  anyone  may 
assign  and  such  that         V—  V'<Sn. 

To  show  that  V  =  V  we  may  proceed  as  follows : 
.       Assume  7^7',  let  V>V, 

and  denote  the  difference  by  d, 

i.e.,  V-V'  =  d. 

We  have  seen  that  by  increasing  the  number  of  divi- 
sions in  the  altitude  h  we  can  make  S  less  than  any 
assigned  quantity,  therefore  less  than  d. 

Hence,  V-V'<d. 

This  contradicts  the  statement 

V-V'~d. 

Therefore  the  assumption  V^  V  is  wrong  and  V  —  V. 


BONA  VENTURA  CAVALIERI 


BON A VENTURA     CAVALIERI 

BONAVENTURA  CAVALIERI,  professor  of 
mathematics  at  Bologna,  was  born  at  Milan 
in  1598  and  died  at  Bologna  in  1647.  He 
was  one  of  the  most  influential  mathematicians 
of  his  day.  Through  his  influence  logarithms 
were  introduced  into  Italy.  He  discovered  the 
law  for  the  area  of  a  spherical  triangle  in  terms  of 
its  spherical  excess.  His  chief  claim  to  renown 
rests  on  his  clear  enunciation  in  1629  of  the  prin- 
ciple of  indivisibles.  This  principle,  first  pub- 
lished by  Cavalieri  in  1635,  though  unsound 
philosophically,  is  of  great  mathematical  signifi- 
cance because  it  became  the  progenitor  of  the 
infinitesimal  calculus. 

Cavalieri's  statement  of  the  principle  in  1635 
said  a  line  was  made  up  of  an  infinite  number  of 
points,  each  without  magnitude,  a  surface  of  an 
infinite  number  of  lines,  each  without  breadth, 
and  a  volume  of  an  infinite  number  of  surfaces, 
each  without  thickness.  This  form  of  statement 
was  vigorously  objected  to  by  contemporary 
mathematicians,  and  it  was  later  correctly  stated 
that  the  method  of  indivisibles  rests  on  the  as- 
sumption that  any  magnitude  may  be  divided 
into  an  infinite  number  of  small  quantities  which 
can  be  made  to  bear  any  required  ratios  to  one 
another. 

.  The  method  of  indivisibles  in  turn  grew  out  of 
the  tedious  method  of  exhaustions  used  by  the 
ancient  Greeks.  In  the  eighteenth  century  the 
integral  calculus  replaced  the  method  of  indi- 
visibles. 

One  may  see  how  the  method  of  indivisibles 
was  used  by  Cavalieri  from  two  examples  given 
on  pp.  280  and  281  of  Ball. 


VOLUMES 


287 


302.  Cavalieri's*  theorem.     Let  V  and  V",  Fig.  187, 
be  two  solids  lying  between  parallel  planes  M  and  N, 


Fig.  187 


and  let  the  two  sections  cut  from  V  and  V  by  any  plane 
parallel  to  M  and  N  be  equal. 

By  dividing  the  distance  between  M  and  TV  into  equal 
parts  and  by  drawing  planes  through  the  points  of  division 
parallel  to  N,  the  solids  may  be  divided  into  slices. 

As  the  number  of  planes  between  M  and  N  is  increased, 
the  slices  become  approximately  prismatic  or  cylindrical. 

Any  two  corresponding  slices  are  equal,  since  they  have 
equal  bases  and  altitudes,  §  294. 

Hence  the  sum  of  all  slices  of  V  is  equal  to  the  sum  of 
all  slices  of  V. 

By  increasing  indefinitely  the  number  of  slices,  and 
by  a  process  of  reasoning  similar  to  that  of  §  301,  it  may 
be  seen  that  V=V\ 

This  fact  is  known  as  Cavalieri's  theorem  and  may  be 
stated  as  follows :  If  two  solids  lie  between  two  given  parallel 

*  Bonaventura  Cavalieri  was  born  in  Milan  in  1598  and  died 
in  Bologna  in  1647.  He  gained  a  reputation  through  his  "principle 
of  indivisibles"  in  which  he  asserted  that  lines  were  made  up  of  an 
infinite  number  of  points,  surfaces  of  an  infinite  number  of  fines, 
and  solids  of  an  infinite  number  of  planes.  Though  unscientific, 
Cavalieri's  method  was  used  for  years  as  a  sort  of  integral  cal- 
culus.   See  Cajori,  p.  171,  and  Ball,  pp.  279  and  280. 


288 


THIRD-YEAR  MATHEMATICS 


planes,  having  their  bases  in  these  planes,  and  if  the  sections 
made  by  any  plane  parallel  to  the  given  planes  are  equal, 
then  the  volumes  of  the  solids  are  equal. 

303.  Theorem:    The  volume  of  a  triangular  pyramid  is 
equal  to  one-third  the  product  of  the  base  by  the  altitude.  * 


E        E 


Given  the  triangular  pyramid  A-B  CD,  Fig.  188,  with 
volume  v,  base  b,  and  altitude  h. 

To  prove  that  v  =  — ~— . 

Proof:  On  BCD  as  base  construct  the  triangular 
prism  BCD-E  having  the  altitude  h,  and  AC  as  one  of 
the  lateral  edges. 

Planes  EAD  and  BAD  divide  this  prism  into  three 
triangular  pyramids,  as  shown  in  Fig.  189. 

Then  A-BCD  =  D-EAF     (§301), 

D-EAF^A-EFD 
A-EFD  =  A-B  ED    Why  ? 

Hence  prism  BCD-E  has  been  divided  into  three 
equal  triangular  pyramids. 

.'.    the  pyramid  A-BCD  =  \  of  the  prism  BCD-E , 


OF 


v=\b-h. 


*This  theorem  was  demonstrated  by  Eudoxus  (b.  408  B.C.). 


VOLUMES  289 

304.  Theorem:  The  volume  of  any  pyramid  is  equal 
to  one-third  the  product  of  the  base  by  the  altitude  i.e., 

By   passing   planes   through  the  /lr\\ 

vertex  V,  Fig.  190,  and  the  diagonals  /M\   \\ 

of  the  base,  the  given  pyramid  may  /inJ£~  %\\ 

be  divided  into  triangular  pyramids  A/-:"  //    \*vf|y 

V-AEB,   V-BEC,  etc.     The  sum  ^£j^Nf 

of  the  volumes  of  these  triangular  B               c 

pyramids  is  the  volume  of  the  given  Fig.  190 
pyramid. 

Thus  v  =  ^hb1+%hb2+etc.  =  %h(bi+b2+etc.). 

.'.     v=\hb, 

where  h  is  the  length  of  the  altitude  and  b  the  area  of  the 
base  of  the  pyramid  V- ABODE. 

EXERCISES 

1.  Find  the  volume  of  a  pyramid  whose  base  is  a  square 
10  in.  long  and  whose  altitude  is  12  inches. 

2.  The  edges  of  a  regular  tetraedron  are  3  in.  long.  Find 
the  volume. 

3.  The  base  of  a  regular  pyramid  is  a  square  each  side  of 
which  is  4  feet.     The  pyramid  is  5  ft.  high.     Find  the  volume. 

4.  Find  the  volume  of  a  regular  hexagonal  pyramid,  the 
perimeter  of  whose  base  is  12  inches.  The  pyramid  is  5  in. 
high. 

5.  Originally  the  great  pyramid  of  Cheops  was  480  ft.  9  in. 
high  and  the  side  of  the  square  base  was  764  ft.  long.  Owing 
to  the  removal  of  coating  the  measurements  are  now  460  ft. 
and  746  ft.  respectively.     How  much  stone  has  been  removed  ? 


290  THIRD-YEAR  MATHEMATICS 

6.  The  area  of  the  base  of  a  regular  quadrangular  pyramid 
is  400.  The  altitude  is  10.  Find  the  volume.  How  far  from 
the  vertex  is  a  section  parallel  to  the  base  whose  area  is  100  ? 

7.  The  base  of  a  regular  pyramid  is  a  regular  hexagon  which 
can  be  inscribed  in  a  circle  of  radius  10.  One  of  the  lateral  edges 
of  the  pyramid  is  20.  Find  the  volume  of  the  pyramid. 
(Harvard.) 

8.  Three  edges  of  a  parallelopiped  are  AB,  AC,  and  AD. 
Prove  that  the  plane  BCD  divides  the  parallelopiped  into  two 
solids  whose  volumes  are  in  the  ratio  of  five  to  one. 

In  what  ratio  are  the  volumes  of  the  two  solids  into  which  the 
parallelopiped  is  divided  by  the  plane  bisecting  these  three 
edges  ?     (Harvard.) 

Volume  of  a  Frustum  of  a  Pyramid 

305.  Denote  the  upper  and  lower  base  of  the  frustum 
of  a  pyramid,  Fig.  191,  by  6i  and  b2,  respectively,  and  their 
distances  from  the  vertex  by  fa  and  fa. 

Then        '  £-g     (§260).' 

Denoting  the  altitude  0\02  of  the  frustum  by  h,  we 
have  fa  =  fa  —  h. 

b2  =     -fa2 
•"  •    6i    (fa-hy 
Vfa_  fa 

Vb~h2-h 
.'.     faVb2-hV%  =  faVbi 
.*.     faVb2-faVb\  =  hVb2 

hVb2 


fa  = 

Vb2-Vbi 

Denote  the  volume  of  the  frustum  by  v. 

Since  the  frustum 

A2D,  =  (P-  A2B2C2D2E2)  -  (P  -A&iCiDiEi) , 


VOLUMES  291 

we  have  ^  =  |W2  — iWi  =  iW2— 3&i(^2~ ^) 
=  §62^2  —  \b\h2 + §  hh 

x,  .  .  1hVb20/b2-Vb1)(VK±VW) 

=  JW+^V/620/62+ V' 6i)_ 

=iW+^(i/^2+v/6ii/W 
.-.    y  =  ^(6i+&2+l/^2) 

EXERCISE 

The  stone  cap  of  a  gatepost  is  in  the  form  of  a  regular  square 
pyramid  whose  base  measures  4  in.  on  a  side  and  whose  altitude  is 
15  inches.  If  the  top  of  the  cap  is  cut  off  by  a  plane  parallel  to  its 
base  and  5  in.  above  it,  what  is  the  volume  of  the  piece  cut  off  ? 
(Board.) 

Volume  of  a  Circular  Cone 

306.  Theorem:  The  volume  of  a  circular  cone  is  equal  to 
one-third  the  product  of  the  base  by  the  altitude, 

i.e.,  v  =  J)'h.  fa   t 

Proof  (indirect  method) :  ij  j  \\ 

1.  Suppose  v<\bh  Ml  !  \% 
and  that                    v  =  \Bh,  where  B  <  b.    A  fflfr  '    T m 

Inscribe  a  pyramid,  Fig.  192,  whose      ^<____J^X 
base  B'>B.  FlG   192 

Then  \B'h>\Bh. 

This  means  that  the  volume  of  the  inscribed  pyramid 
is  greater  than  the  volume  of  the  cone. 

This  is  impossible,  and  v  is  not  less  than  \bh. 

2.  Similarly  we  may  show  that  v  is  not  greater  than  \bh. 

3.  Hence  v  =  A>  •  h. 


292  THIRD-YEAR  MATHEMATICS 

EXERCISE 

A  right  triangle  is  revolved  about  one  leg.  Show  that  the 
volume  of  the  cone  thus  generated  is  equal  to  the  product  of  the 
area  of  the  triangle  and  the  circumference  of  the  circle  traced 
by  the  point  of  intersection  of  the  medians.     (Harvard.) 

Volume  of  a  Frustum  of  a  Cone 

307.  The  formula  giving  the  volume  of  the  frustum  of 
a  cone  is  obtained  in  the  same  way  as  the  formula  for 
the  volume  of  the  frustum  of  a  pyramid. 

Hence,  v  =  \h(bi+b2+V  bib2) - 

308.  To  find  the  volume  of  a  frustum  of  a  cone  of  revo- 
lution let  bi  =  7rn2  and  b2=irr22. 

Then  Vb&2 = V-n-r^irr^  =  irnr2. 

Substituting  these  values  in  §  307, 

v=lirh(ri2+r2*+rir2). 

EXERCISES 

1.  The  bases  of  a  frustum  of  a  pyramid  are  regular  hexagons 
whose  sides  are  8  in.  and  4  in.  respectively.  The  altitude  of  the 
frustum  is  3  feet.     Find  the  volume. 

2.  The  radii  of  the  bases  of  a  frustum  of  a  cone  of  revolution 
are  4  in.  and  5  inches.  The  frustum  is  12  in.  high.  Find  the 
volume. 

3.  A  conical  heap  of  grain  is  4  ft.  high  and  has  a  circular 
base  whose  radius  is  5  feet.  How  high  must  a  bin  be  whose 
base  is  4  ft.  square  to  contain  the  grain  ? 

4.  Find  the  number  of  bushels  of  wheat  contained  in  conical 
heap  thrown  into  a  corner  of  a  bin,  the  highest  point  of  the  heap 
being  4  ft.  and  the  radius  of  the  circular  base  being  6 . 5  feet  ? 

A  bushel  contains  2,150  cubic  inches. 

5.  A  cone  is  12  in.  high  and  the  area  of  the  base  is  15  square 
inches.     Find  the  volume. 


VOLUMES 


293 


6.  The  height  of  the  frustum  of  a  cone  is  6  in.  and  the  radii 
of  the  bases  are  4  in.  and  8  in.,  respectively. 

Find  the  volume. 

7.  What  must  be  the  depth  of  a  pail 
that  is  18  in.  aeross  the  top  and  10  in. 
across  the  bottom  in  order  that  it  may 
hold  5,280  cubic  inches?    (*-V.)    (Yale.) 

8.  A  pyramid  is  6  in.  high.  The  area 
of  its  base  is  324  square  inches.  Find  the 
volume  of  the  frustum  cut  off  by  a  plane 
4  in.  from  the  base.  j,       igo 

9.  Find  the  volume  of  a  grain  tank, 

Fig.  193,  10  ft.  high  and  9  ft.  in  diameter,  the  height  of  the  roof 
being  3  feet. 

Volume  of  a  Sphere 

309.  Theorem:   The  volume  of  a  sphere  of  radius  r  is 

Let  ACB,  Fig.  194,  be  a  hemisphere,  and  let  DF  be  a 
right  cylinder  whose  circular  base,  DE,  is  equal  to  the 


i 

c 
111     Q     \\2 

k£ 

G(, 

^r?~~. 

r,~ . 

-^ 

\F 

K 

m\ 

1 

L  H   J 

-jf 

ip  z 

/     ^ 

f  B 

d\ 

\e/ 

/ 

Fig.  194 

circle  A B  and  whose  altitude  is  equal  to  the  radius,  r, 
of  the  sphere. 

Suppose  a  cone  H  —  GF  be  cut  from  the  cylinder 
leaving  the  solid,  GDEFHG. 

Pass  a  plane  parallel  to  plane  Z  and  at  a  distance  x 
from  Z,  and  let  KL  and  MNOP  be  the  sections  of  ACB 
and  GDEFH,  respectively. 


294  THIRD-YEAR  MATHEMATICS 


Show  that  RL  =  Vr2  -  x2. 

au       ..  UO    HU        UO    x 

Show  that  YF  =  Hf,0V~T  =  V 

:.    OU=x. 

Show  that  the  area  of  KL  =  -rr{r2  —  x2). 

Show   that   the   area   of   the    circular   ring   MNOP 

=  7rr2  —  ttX2  =  ir{r2  —  X2) . 

.'.  the  hemisphere  ACB  is  equal  to  the  solid  GDEFHG, 
§302. 

Since  GDEFHG  =  (DF)  -(H-  GF) 

=  ttt2  •  r  —  \-rrr2  •  r  —  fir3, 

it  follows  that  the  hemisphere  ACB  =  \ttis'. 

.'.  the  volume  of  the  sphere  is  given  by  the  formula 

y  =  3'irr3. 

EXERCISES 

1.  •  Find  the  weight  of  a  cast-iron  sphere  4  in.  in  diameter. 
Cast-iron  weighs  .26  lb.  per  cubic  inch. 

2.  Find  the  volume  of  a  sphere  4  in.  in  diameter. 

3.  Find  the  volume  of  metal  in  a  spherical  shell  \  in.  thick 
whose  external  diameter  is  4  inches. 

4.  The  area  of  a  spherical  surface  is  6  square  inches.  Find 
the  volume  of  the  sphere. 

6.  A  bar  of  metal  of  the  form  of  a  rectangular  parallelopiped 
12  X  8  X  4  in.  is  to  be  melted  and  cast  into  a  spherical  ball.  What 
is  the  radius  of  the  ball  ? 

No  allowance  is  to  be  made  for  waste. 

6.  Prove  that  the  volumes  of  two  spheres  are  to  each  other 
as  the  cubes  of  the  radii. 

7.  Regarding  the  earth  and  the  sun  as  spheres  of  radii 
4,000  mi.  and  860,000  mi.,  respectively,  compare  their  volumes. 


VOLUMES  295 

8.  A  rifle  shell  has  the  shape  of  a  cylinder  surmounted  by  a 
hemispherical  cap.  The  total  length  of  the  shell  is  four  times  its 
diameter.  Compare  the  surfaces  and  also  the  volumes  of  the 
cylindrical  and  the  spherical  portions.     (Sheffield.) 

9.  Find  the  volume  and  surface  of  a  sphere  inscribed  in  a 
cube  whose  diagonal  is  6V7 3.     (Yale.) 

10.  A  sphere  is  inscribed  in  a  cube.  Find  the  ratio  of  the 
radius  of  the  sphere  to  the  edge  of  the  cube. 

11.  What  percentage  of  the  volume  of  a  sphere  is  contained 
in  the  inscribed  cube  ?     (Harvard.) 

12.  A  regular  octaedron  has  an  edge  a.  Find  the  volume  of 
the  inscribed  sphere.     (Harvard.) 

13.  In  a  semicircle  of  radius  a  is  inscribed  a  right  triangle 
one  of  whose  acute  angles  is  30°,  the  hypotenuse  of  the  triangle 
being  the  diameter  of  the  circle.  The  figure  is  revolved  about 
the  diameter  as  an  axis.  Find  the  ratio  of  the  volumes  generated 
by  the  triangle  and  the  semicircle.     (Yale.) 

14.  The  inside  of  a  glass  is  in  the  form  of  a  cone  whose  verti- 
cal angle  is  60°,  and  whose  base  is  2  in.  across.  The  glass  is 
filled  with  water  and  the  largest  sphere  that  can  be  immersed  is 
placed  in  the  glass.  How  much  water  remains  in  the  glass? 
(Yale.) 

15.  A  hemisphere  and  a  right  circular  cone  have  the  same 
base,  and  the  areas  of  their  curved  surfaces  are  equal.  Find  the 
ratio  of  their  volumes.     (Harvard.) 

Volume  of  a  Spherical  Segment 

310.  Spherical  segment.    The        Xjg     ^N 

portion  of  a  sphere  included  between      S~ ^ 

two   parallel  planes  intersecting  a      ^ 

.  Spherical  Segment 

sphere    is    a    spherical   segment,  of  two  bases 

Fig.  195.  Fig.  195 


296 


THIRD-YEAR  MATHEMATICS 


The  perpendicular  between  the  planes 
is  the  altitude,  the  sections  of  the  sphere 
made  by  the  planes  are  the  bases  of  the 
segment. 

If  one  of  the  planes  is  tangent  to  the 
sphere  the  segment  has  only  one  base, 
Fig.  196. 


Spherical  Segment 
of  one  base 

Fig.  196 


311.  Theorem:    The  volume  of  a  spherical  segment  of 
one  base  is  given  by  the  formula 

V=l?i2Tr(3r-h), 

where  r  is  the  radius  of  the  sphere  and  h  the  altitude  of  the 
segment. 

Proof:   According  to  §§  302,  309,  the  segment  KCL, 
Fig.  197,  is  equal  to  the  solid  MPFONG,  which  is  the 


Fig.  197 

difference  between  the  cylinder  M F  and  the  frustum  of  a 
cone,  GNOF. 

The  radii  of  the  bases  of  the  frustum  are  TF  =  r  and 
UO  =  x  =  r-h. 

.-.     MF  =  7rr2h 
and    GNOF  =  lirh[r2+(r-h)2+r(r-h)] 
=  %irh(3r2-3rh+h2) 
.'.  KCL  =  MF-GNOF  =  Trr2h-l7rh(3r2-3rh+h2) 
=  i*h(3r2-3r2+3rh-h2) 
=  \nh(3rh-h2) 
:.    V=lTTh2(Sr-h) 


VOLUMES  297 

312.  Theorem:  The  volume  of  a  spherical  segment  of 
two  bases  is  given  by  the  formula 

V=l(Trrf+irr2*)+-~. 

Proof:  The  volume  of  a  segment  of  two  bases  is  equal 
to  the  difference  of  two  segments  having  one  base. 

Denoting  the  altitude  of  the  segment  of  two  bases 
by  h  and  the  altitudes  of  the  segments  of  one  base  by 
hi  and  h2,  respectively,  we  have  h  =  hi  —  h2. 

.-.    v  =  ^7rhi2(Sr-hi)-^Trh22(Sr-h2) 

=  irrhi2  —  ^-rrhi3  —  itrh22 + ^vh2s 

=  7rr(hi2-h22)-^7r(hiz-h2s) 

=  irrihi-h2)(hi+h2)-^7r(hi-h2)(hi2+hih2+h22) 

«  (hi-h2)[r(hi+h2)-\{hi2+hih2+h22)} 

=  irh[rhi+rh  - §  (h2  -  2hih2+h22+3hh2)] 

=  7rh[rhi+rh2-^(h2+3hih2)] 

I  h2 

=  -rrh  (  rhi  +rh2——  —  hih2 

h2         r2      _...,,,_  _x  h         n 


Show  that  —  =  « =-  and  that  —  = 


r2     2r  —  h2  n     2r  —  hi} 

.-.     2rh2-h22  =  r22 
and  2rhi  —  hi2  =  n2. 


Adding,  2rh+2rh2-  (h2+h22)  =  n2+r22 

n2+r22  ,  h2+h22 


rhi-\-rh 


Jn2+r22  ,  h2+h22    h2     ,,\ 
..     ,.^__+_ ~-hih2) 

t/ri?fr22  ,  h2-\-2hih2     h2     2hih2\ 


777 

2 


K-J(«rf+*i*)+^ 


298 


THIRD-YEAR  MATHEMATICS 


313.  Spherical  cone.  A  spherical  cone,  Fig.  198,  is 
generated  by  revolving  a  circular  sector,  ABC,  Fig.  199, 
about  its  bounding  radius,  BC,  as  an  axis. 


314.  Theorem:   The  volume  of  a  spherical  cone  is  given 


by  the  formula. 


v=*pr2h. 


Proof:  ABDC=ABD+ACD 

=  ±7rx*(r-h)+lTrh2(Sr-h) 
=  ^[r2-(r-h)2](r-h)+^7rh2(3r-h) 
=  %Tr(2rh-h2)(r-h)+^h2(3r-h) 
=  ±7rh(2r2-hr-2rh+h2+3rh-h2) 
=  \irh-2r2 
.'.    v  =  \-nr2h. 

315.  Spherical  sector. 

The  portion  of  a  sphere 
generated  by  revolving 
a  circular  sector  ABC, 
Fig.  200,  about  a 
diameter  of  its  circle 
is  a  spherical  sector, 
Fig.  201. 

316.  Theorem:    The  volume  of  a   spherical  sector  is 
given  by  the  formula 

v=\-nr2h. 


B 

Fig.  201 


VOLUMES  299 

Proof:  A  spherical  sector  is  the  difference  between 
two  spherical  cones.  Denote  their  volumes  by  vi  and  v2 
respectively. 

Then  vi  =  %-n-r2hi 

V 


v2  =  1 7rr2/l2 


v\  —  V2  =  §  Trr2(hi  —  hz) 


or  v  =  §Trr2/i. 


EXERCISES 

1.  The  distance  of  a  plane  from  the  center  of  a  sphere  is  one- 
third  the  radius  of  the  sphere.  Find  the  ratio  of  the  volumes  of 
the  two  solids  into  which  the  sphere  is  divided  by  this  plane. 
(Harvard.) 

2.  In  a  certain  sphere  there  are  as  many  square  feet  in  the 
surface  as  there  are  cubic  feet  in  the  volume.  Find  the  radius 
and  determine  the. area  of  the  segment  of  this  spherical  surface 
cut  off  by  a  plane  perpendicular  to  the  radius  at  its  middle  point. 

$3.  How  large  a  hole  must  be  bored  through  a  sphere  6  in. 
in  diameter  to  remove  one-half  of  the  sphere  ? 

The  part  cut  from  the  sphere  consists  of  a  cylinder,  C,  and  two 
spherical  segments,  S. 

27rr3 
Show  that  2S  =  — 5—  (2+ cos  x  — 2  cos2  z  — cos3  x) 

o 

and  that  C  =  2-irr3  (cos  x  —  cos3  x) . 

$4.  The  diameter  of  a  sphere  is  10  inches.  If  a  cylindrical 
hole  of  5  in.  in  diameter  is  bored  through  the  sphere,  what  is 
the  volume  of  the  remaining  solid?  It  is  assumed  that  the 
center  of  the  sphere  lies  on  the  axis  of  the  cylinder. 

5.  The  curved  surface  of  a  spherical  segment  of  one  base  is 
25V  and  the  altitude  is  3.    Find  the  volume. 


300  THIRD-YEAR  MATHEMATICS 

Summary 

317.  The  chapter  has  taught  the  meaning  of  the  fol- 
lowing terms : 

unit  of  volume,  volume 

inscribed  prism,  pyramid,  and  frustum  of  a  pyramid 

circumscribed  prism,  pyramid,  and  frustum  of  a  pyramid 

tangent  plane 

spherical  segment,  cone,  and  sector 

318.  The  following  theorems  have  been  studied : 

1.  The  plane  passed  through  two  diagonally  opposite 
edges  of  a  right  parallelopiped  divides  the  parallelopiped  into 
two  equal  triangular  right  prisms. 

2.  An  oblique  prism  is  equal  to  a  right  prism  whose  base 
is  equal  to  a  right  section  of  the  oblique  prism  and  whose 
altitude  is  equal  to  the  lateral  edge  of  the  oblique  prism. 

3.  The  plane  passed  through  two  diagonally  opposite 
edges  of  any  parallelopiped  divides  the  parallelopiped  into 
two  equal  triangular  prisms. 

4.  Prisms  having  equal  bases  and  altitudes  are  equal. 

5.  The  volumes  of  two  similar  cylinders  of  revolution  are 
to  each  other  as  the  cubes  of  the  altitudes,  or  as  the  cubes  of  the 
radii  of  the  bases. 

6.  If  two  pyramids  have  equal  bases  and  equal  altitudes, 
sections  made  by  planes  parallel  to  the  bases  and  at  equal 
distances  from  the  vertices  are  equal. 

7.  If  two  triangular  pyramids  have  equal  bases  and 
altitudes,  they  are  equal. 

8.  If  two  solids  lie  between  two  given  parallel  planes, 
having  their  bases  in  these  planes,  and  if  the  sections  made 
by  any  plane  parallel  to  the  given  planes  are  equal,  then  the 
volumes  of  the  solids  are  equal. 


VOLUMES  301 

319.  The  following  is  a  summary  of  the  formulas  in 
this  chapter: 

Rectangular  parallelopiped v  =  aXbXc 

v  =  bxh  (6  =  base) 

Cube v=e* 

Triangular  right  prism v  =  bxh 

Right  parallelopiped v  =  bXh 

Oblique  parallelopiped v=bXh 

Triangular  prism v=bxh 

Prism v  =  bxh 

Cylinder v  =  bXh 

Cylinder  of  revolution v=tn2h 

Pyramid v  =  \bXh 

Frustum  of  a  pyramid v=bi(bi+bi+V  bib2) 

Cone v=\hXb 

Frustum  of  a  cone  of  revolution v^gvftfcH-itf+ivO 

Sphere i/= -Jirr3 

Spherical  segment  of  one  base v=*trh2(Zr-h) 

Spherical  segment  of  two  bases v=^(trri2+Tn22)+^w- 

2  6 

Spherical  cone v=\trr2h 

3 

Spherical  sector v=%r2h 


CHAPTER  XIV 


POLYEDRAL  ANGLES.    TETRAEDRONS. 
SPHERICAL  POLYGONS 

Polyedral  Angles 

320.  Polyedral  angle.  If  a  line,  AB,  Fig.  202,  moves 
with  one  endpoint  fixed  at  A  and  always  touching  a  con- 
vex polygon,  CDEFG,  whose  plane  does  not  contain  A, 
it  generates  a  convex  polyedral  angle. 


The  fixed  point  A  is  the  vertex,  the  bounding  planes 
CAD,  DAE,  etc.,  are  the  faces,  the  lines  AC,  AD,  etc., 
are  the  edges,  A  CAD,  DAE,  etc.,  are  the  face  angles  of 
the  polyedral  angle. 

321.  Triedral  angle.  A  polyedral  angle  having  three 
faces,  as  (1),  Fig.    202,  is  a  triedral 

angle.  ° 

Point  out  several  triedral  angles  in 
the  classroom. 

322.  Theorem:    The  sum  of  two  \d 

face  angles  of  a  triedral  angle  is  greater 
than  the  third. 


Fig.  203 


Given  the  triedral  angle  O-ABC,  Fig.  203. 
To  prove  that  IAOB+  Z  BOO  Z.AOC. 
302 


POLYEDRAL  ANGLES  303 

Proof:  The  theorem  is  easily  proved  for  a  triedral 
angle  having  equal  face  angles. 

Assume  that  the  face  angles  are  not  equal  and  that 
ZAOC  is  the  greatest  face  angle. 

In  the  plane  AOC  draw  OD,  making  ZAOD  =  ZAOB. 

Lay  off  OD'=OB'. 

Pass  a  plane  through  B'  and  D',  cutting  the  faces  in 
lines  A'B\  B'C\  and  C'A',  respectively. 

Prove  that       AA'OB'  &  AA'OD'. 

.'.A'B'  =  A'D'. 
Show  that  A'B'+B'OA'C. 
Subtracting,  B'OD'C. 

.*.  ZB'OO  ZD'OC. 
.'.  /.A'OB'+AB'OOAA'OD'+AD'OC, 
or  ZAOB+  /.BOO  ZAOC. 

EXERCISES 

1.  Show  that  the  difference  of  two  face  angles  of  a  triedral 
angle  is  less  than  the  third. 

2.  Show  that  any  face  angle  of  a  polyedral  angle  is  less 
than  the  sum  of  the  other  face  angles. 

3.  Show  that  the  three  planes  bisecting  a  triedral  angle 
intersect  in  a  straight  line. 

323.  Spherical  polygon.  Let  the  faces  of  the  polyedral 
angle  O-ABCD,   Fig.   204,   intersect   the   surface   of   a 


Fig.  204 


304 


THIRD-YEAR  MATHEMATICS 


sphere  whose  center  is  0  in  the  great  circle  arcs  AB,  BC, 
CD,  and  DA.  The  figure  ABCD  on  the  surface  of  the 
sphere  is  a  spherical  polygon. 

Thus  a  spherical  polygon  is 
the  section  of  a  spherical  surface 
made  by  a  convex  polyedral 
angle  whose  vertex  is  at  the  center 
of  the  sphere.  To  every  polyedral 
angle  at  the  center  of  the  sphere 
corresponds  a  spherical  polygon. 

A  spherical  polygon  of  three 
sides  is  a  spherical  triangle, 
Fig.  205.* 

The  bounding  arcs,  AB,  BC,  etc.,  are  the  sides  of  the 
spherical  polygon.  The  points  of  intersection  of  the  sides 
are  the  vertices  of  the  polygon. 


Fig.  205 


EXERCISES 

1.  The  sides  of  a  spherical  polygon 
are  usually  measured  in  degrees.  Show 
that  the  sides  of  a  spherical  polygon  have 
the  same  measure  as  the  face  angles  of 
the  corresponding  polyedral  angle  at  the 
center  of  the  sphere,  Fig.  206. 

*  The  properties  of  spherical  triangles  are  applied  in  the  solution 
of  problems  in  astronomy,  navigation,  and  geography.  In  fact, 
spherical  geometry  was  first  developed  by  astronomers.  The  follow- 
ing are  some  of  the  interesting  applications: 

1.  To  determine  the  position  of  an  observer  on  the  surface  of  the 
earth,  i.e.,  his  latitude  and  longitude. 

2.  To  find  the  distance  between  two  places  and  the  bearing  of 
each  from  the  other  when  their  latitudes  and  longitudes  are  known. 

3.  To  determine  the  position  of  a  star. 

4.  To  determine  the  time  of  the  day  at  a  place  on  the  surface 
of  the  earth. 

5.  To  determine  the  course  of  a  ship. 


POLYEDRAL  ANGLES  305 

2.  Two  sides  of  a  spherical  triangle  are  88°  and  70°.  What 
are  the  limits  for  the  third  side  ? 

Exercise  2  indicates  how  some  properties  of  spherical  polygons 
may  be  inferred  from  a  study  of  polyedral  angles.  , 

3.  Show  that  the  sum  of  two  sides  of  a  spherical  triangle  is 
greater  than  the  third  side,  Fig.  206. 

4.  The  shortest  line  that  can  be  drawn  between  two  given  points 
on  the  surface  of  a  sphere  is  the  minor  arc  of  the  great  circle  which 
passes  through  the  two  points.    Prove. 

Proof:  1.  Let  A  and  B,  Fig.  207,  be  the  two  given  points  and 
let  ACB  be  the  minor  arc  of  a  great  circle  joining  A  and  B. 

Let  C  be  any  point  onAB.    With  A  and  B  as  centers  and  radii 
equal  to  AC  and  BC,  respectively,  draw 
two  small  circles  meeting  at  C.  A 

Let  D  be  any  point  on  circle  A, 
not  point  C,  and  draw  the  arcs  of  great 
circles  AD  and  DB. 

Then  AD+DB>AB,  §  323,  exer- 
cise 3.        ^  ^ 
But    AD           =  AC, 

DB>CB. 

Thus  D  lies  outside  of  circle  B. 

.'.  circles  A  and  B  are  tangent  to 
each  other  at  C. 

2.  Let  AEFB  be  any  line  joining 
A  and  B  on  the  surface  of  the  sphere  and  not  passing  through  C. 
Then  line  AEFB  meets  circles  A  and  B  in  two  distinct  points,  E  and 
F.     Why? 

Whatever  may  be  the  form  of  AE,  an  equal  line  can  be  drawn 
from  A  to  C;  and  whatever  may  be  the  form  of  BF,  an  equal  line 
can  be  drawn  from  B  to  C.     Why  ? 

Hence  it  is  possible  to  draw  a  line  from  A  to  B  passing  through 
C  and  equal  to  AE+FB. 

Since  AE+FB<AE+EF+FB,  it  is  always  possible  to  draw 
a  line  from  A  to  B  passing  through  C  and  shorter  than  AEFB. 

Thus  the  shortest  line  from  A  to  B  passes  through  C. 

Since  C  is  any  point  on  AB,  the  shortest  line  from  A  to  B  passes 
through  every  point  of  AB  and  therefore  is  the  arc  AB. 


Fig.  207 


306  THIRD-YEAR  MATHEMATICS 

324.  Theorem:  The  sum  of  the  face  angles  of  a  convex 
polyedral  angle  is  less  than  four  right  angles. 

Given   the   convex  polyedral  0 

angle  O-AB  CDE,  Fig.  208.  JL 

To  prove  that  /  \  \\ 

ZAOB+ZBOC+ZCOD+  /    \  \\ 

etc.,  <4  R.A.  aJz^/^ZS^? 

Proof:  Let  AB  CDE  be  a  section         \/^    X\/ 
of  the  polyedral  angle  made  by  a  ib  c\ 

plane  cutting  all  the  edges.  Fig.  208 

From  any  pfoint  0'  within 
ABODE  draw  lines  to  the  vertices  A,  B,  C,  etc. 

In  the  triedral  angle  B-AOC, 

Z  ABO+  Z  OBC  >  Z  ABC,  §  322. 

In  the  triedral  angle  C-BOD, 

ZBCO+ZOCD>  A  BCD,  etc. 

Adding, 

Z  ABO+  Z  CBC+  Z  BCO+  Z  (XLD+etc, 

>  ZABC+ZBCD+etc, 

i.e.,  the  sum  of  the  base  angles  of  the  triangles  with  vertex 
0  is  greater  than  the  sum  of  the  base  angles  of  the  triangles 
with  vertex  0'. 

But  the  sum  of  all  angles  of  the  triangles  with  vertex  0  is 
equal  to  the  sum  of  all  angles  of  the  triangles  with  vertex  0'. 

: .  by  subtracting  unequals  from  equals  we  have  the  sum 
of  the  face  angles  at  Oless  than  the  sum  of  the  angles  about  0'. 

In  symbols  this  may  be  stated  as  follows : 

ZAOB+  ZBOC+  ZCOD+etc,  <  ZAO'B+ZBO'C 
+  ZCO,D+etc. 

Since 

ZAOfB+Z  BO'C+  Z  CO'D+etc.  =  4  R.A. 
,\  ZAOB+  ZBOC+  ZCOD+  etc. <4R.A. 


POLYEDRAL  ANGLES  307 

EXERCISE 

Prove  that  the  sum  of  the  sides  of  any  convex  spherical  polygon 
is  less  than  360°,  Fig.  209. 

325.  Number  of  regular  poly- 
edrons.  In  §  245  five  regular 
polyedrons  were  shown.  The 
theorem  in  §  324  may  be  used  to 
prove  that  there  are  no  other  kinds 
of  convex  regular  polyedrons. 

For  the  faces  of  a  regular 
polyedron  are  all  regular  poly-  fig.  209 

gons  such  as  equilateral  triangles, 

squares,  etc.,  and  the  sum  of  the  face  angles  of  any 
polyedral  angle  of  the  polyedron  must  be  less  than  360°. 
Why? 

Show  that  three,  four,  or  five  equilateral  triangles, 
but  not  six  or  more,  may  be  placed  so  as  to  form  a  polyedral 
angle.  Hence  no  polyedron  can  be  formed  with  six  or 
more  equilateral  triangles  at  the  vertex.  The  tetraedron 
has  three  equilateral  triangles  at  one  vertex,  the  octaedron 
has  four,  and  the  icosaedron  has  five. 

Show  that  three  squares  may  be  placed  so  as  to  form  a 
polyedral  angle,  but  not  four  or  more.  Hence  no  polye- 
dron can  be  formed  with  four  or  more  squares  at  a  vertex. 
The  cube  has  three  squares  at  one  vertex. 

Show  that  three  regular  pentagons  may  be  placed  so  as 
to  form  a  polyedral  angle,  but  not  four  or  more.  Hence 
the  dodecaedron  is  the  only  polyedron  whose  faces  are 
regular  pentagons. 

Show  that  it  is  impossible  to  form  a  regular  polyedron 
having  six  or  more  regular  polygons  at  one  vertex.* 

*  The  regular  solids  were  studied  so  extensively  by  Plato  and  his 
school  that  they  have  received  the  name  of  " Platonic  figures." 


308 


THIRD-YEAR  MATHEMATICS 


Tetraedrons 

326.  Theorem :  Two  tetraedrons  having  a  triedral  angle 
of  one  equal  to  a  triedral  angle  of  the  other  are  to  each  other  as 
the  products  of  the  edges  including  the  equal  triedral  angles. 

Given  the  tetraedrons  T-ABC  and  T'-A'B'C, 
Fig.  210,  with  the  triedral  angle  at  T  equal  to  the  triedral 


Fig.  210 
angle  at  T'  and  having  the  volumes  equal  to  V  and  V 
respectively  TAXTBXTC 

To  prove  that  y,  =  rA,xrB,xrc, • 

Proof:     Place  T'-A'B'C'  on  T-ABC,  making  triedral 
angle  T'  coincide  with  triedral  angle  T. 
Draw  C'P'  and  CPLATB. 

V_      \XABTXCP 

V 


Then 


ABT  v  CP 


WaTB'TXC'P'    A'B'T^C'P' 


§303. 


Since  triangles  ABT  and  A'B'T  have  one  angle  equal, 
ABT      TAXTB 


A'B'T 
show  that 
CP 


TA'XTB' 

TC 
TC 


or 


CP' 
By  substitution, 

V^^  TAXTB 

V'~ 

V        TAXTBXTC 

V 


TC 
TA'XTB'^TC" 


T'A'XT'B'XT'C 


POLYEDRAL  ANGLES 


309 


327.  Similar  polyedrons.  Two  polyedrons  are  similar 
if  their  faces  are  similar  each  to  each  and  similarly  placed 
and  if  the  corresponding  polyedral  angles  are  equal. 

328.  Theorem:  Two  similar  tetraedrons  are  to  each 
other  as  the  cubes  of  the  corresponding  edges. 

V  TAXTBXTC 

V  TA'XT'B'XT'C" 
TA       TB       TC 


§326. 


Show  that 


T'A' 
V 
V 


T'B' 
TA 


T'C 
TA 


/  ^\  rni  A  /  ^ 


TA       TA' 


T'A'"T'A'"T'A'     TA 


/3 


EXERCISE 

A  pyramid,  the  area  of  whose  base  is  36  sq.  ft.,  contains  -g1^ 
of  the  volume  of  a  similar  pyramid  whose  altitude  is  9  feet.  Find 
the  volume  of  each  pyramid. 

329.  To  construct  a  sphere  through  four  given  points  not 
all  in  the  same  plane. 

Given  the  four  points  A,  B,  C,  and 
D,  Fig.  211,  not  all  in  the  same  plane. 

To  construct  a  sphere  passing 
through  A,  B,  C,  and  D. 

Construction:  Draw  AB,  AC,  AD, 

BC,  CD,  and  DB  forming  the  tetra- 
edron  A-BCD. 

Bisect  CD  at  E. 

Draw  plane  FEG  perpendicular  to  CD  at  E  and  inter- 
secting planes  CAD  and  CBD  in  lines  EF  and  EG  respec- 
tively. 

Show  that  EF  passes  through  the  center  F  of  the  circle 
circumscribed  about  ACAD  and  that  EG  passes  through 
the  center  G  of  the  circle  circumscribed  about  &CBD. 


310 


THIRD-YEAR  MATHEMATICS 


Draw  FI±  plane  CAD  and  GH±  plane  CBD. 

Since  CD _L  plane  FEG,  it  follows  that  planes  CAD  and 
Cj5D  are  perpendicular  to  plane  FEG. 
Why? 

Show  that  FI  and  (?#  lie  in  plane 
FEG,  §  551. 

Show  that  FI  and  (x#  are  not 
parallel. 

Denoting  the  point  of  intersection 
of  FI  and  GH  by  0,  show  that  0  is 
equidistant  from  A,  B,  C,  and  D. 

Therefore  a  sphere  with  0  as  center  and  radius  OB 
passes  through  A,  B,  C,  and  D. 

330.  To  inscribe  a  sphere  in  a  given  tetraedron. 

Given  the  tetraedron  A- BCD,  Fig.  212. 

To  construct  a  sphere  tangent  to  all  faces  of  A- BCD. 


Fig.  211 


Construction:  Draw  plane  BOD  bisecting  the  diedral 
angle  BD. 

Draw  plane  BOC  bisecting  diedral  angle  BC. 

These  planes  must  meet  in  a  line,  as  BO,  since  they 
have  point  B  in  common. 


POLYEDRAL  ANGLES 


311 


Draw  plane  CO  A  bisecting  the  diedral  angle  AC. 
This  plane  will  meet  the  line  of  intersection  of  planes  BOD 
and  BOC  in  point  0. 

Show  that  0  is  equidistant  from  the  four  faces  of  the 
tetraedron  A-BCD. 

Hence  a  sphere  with  0  as  center  and  radius  equal  to 
the  perpendicular  from  0  to  one  of  the  faces  will  be  tangent 
to  all  faces. 

1 331.  To  determine  the  diameter  of  a  given  material  sphere. 


Fig.  213 

Given  a  material  sphere,  0,  Fig.  213. 
To  find  its  diameter. 

Construction:  With  P,  a  point  on  the  surface  of  the 
sphere,  as  a  pole,  describe  the  circle  ABC. 

Let  A,  B,  and  C  be  three  points  on  this  circle. 

Construct  triangle  A  '.B'C'congruent  to  the  triangle  ABC. 

Circumscribe  a  circle  about  AA'B'C,  and  let  D'  be 
the  center  of  this  circle. 

Draw  D"A"  equal  to  the  radius  D'A'. 

Through  D"  draw  a  line  P"Q"  perpendicular  to  D" A" . 

From  A"  lay  off  A"P"  equal  to  AP. 

At  A"  erect  A"Q"  perpendicular  to  A"P". 

Then  P"Q"  is  the  required  diameter  of  the  given  sphere. 

The  proof  is  left  to  the  student. 


312  THIRD-YEAR  MATHEMATICS 

Spherical  Angles 

332.  Spherical  angle.  Two  intersecting  curves,  C 
and  C,  Fig.  214,  are  said  to  form  an  angle.  The  angle 
formed  by  two  intersecting  curves  is  the  angle  made  by  the 
tangents  to  the  curves  at  the  common  point,  as  Z  TOT'. 


Fig.  214  Fig.  215 

The  angle  formed  by  two  intersecting  arcs  of  great 
circles  is  a  spherical  angle,  as  TOT',  Fig.  215.  The  point 
of  intersection,  0,  is  the  vertex  and  the  arcs  OA  and  OB 
are  the  sides  of  the  spherical  angle. 

333.  Measure  of  a  spherical  angle.  Draw  AB, 
Fig.  215,  an  arc  of  a  great  circle  with  0  as  a  pole  and 
terminated  by  the  sides  of  the  spherical  angle  AOB. 

Draw  the  radii  CO,  CA,  and  CB. 

Show  that  OC  is  perpendicular  to  OT  and  CA, 

.*.    OTWCA. 
Similarly,  OT'WCB. 

...     ZT0T'=ZACB. 

But  Z.ACB  has  the  same  measure  as  arc  AB. 
:.  /.TOT',  or  spherical  angle  AOB,  is  measured  by 
arc  AB. 


POLYEDRAL  ANGLES 


313 


This  proves  the  following  theorem: 

A  spherical  angle  is  measured  by  the  arc  of  a  great  circle 
having  the  vertex  as  pole,  and  included  between  the  sides, 
produced  if  necessary. 


EXERCISES 

1.  Prove  that  a  spherical  angle  is  equal  to  the  diedral  angle 
formed  by  the  planes  of  the  sides. 

2.  The  angles  formed  by  the  sides  of  a  spherical  triangle  are 
respectively  equal  to  the  diedral  angles  of  the  corresponding 
triedral  angle.     Prove. 

334.  Right  spherical  angle.  When  the  planes  of  the 
sides  of  an  angle  are  perpendicular  to  each  other,  a  right 
spherical  angle  is  formed. 

335.  Classification  of  spherical  triangles.  The  terms 
isosceles,  equilateral,  and  scalene  have  the  same  mean- 
ing for  spherical  triangles  as  for  plane  triangles.  A 
spherical  triangle  is  right,  birectangular,  or  trirectangular, 


Fig.  216 


according   as  it   has   one,  two,  or  three  right  angles, 
Fig.  216. 


314 


THIRD-YEAR  MATHEMATICS 


Polar  Spherical  Triangles 

336.  Polar  triangles.  Let  AABC,  Fig.  217, 
given  triangle.  Draw  three  great  circle  arcs  as 
B'C,  and  CA',  having  as  poles 
C,  A,  and  B,  respectively.  Any 
two  of  these  circle  arcs,  if  far 
enough  extended,  have  two  points 
of  intersection.  Let  C  be  that 
point  of  intersection  of  arcs  A'C 
and  B'C  which  is  nearest  to  C, 
let  B'  be  the  point  of  intersection 
of  arcs  A'B'  and  CB'  which  is 
nearest  to  B,  and  let  A'  be  the 
point  of  intersection  of  arcs  B'A' 
and  CA'  which  is  nearest  to  A. 
polar  triangle  of  AABC. 


be  a 
A'B' 


Fig.  217 


Then  AA'B'C  is  the 


337.  Theorem :  If  a  spherical  triangle  is  the  polar  triangle 
of  another,  then  the  second  is  the  polar  triangle  of  the  first. 

Given  AABC,  Fig.  218,  and 
AA'B'C,  the  polar  of  AABC. 

To  prove  that  AABC  is  the 
polar  of  AA'B'C. 

Proof:  Since  A  is  the  pole  of 
B'C,  it  follows  that  B'  is  a 
quadrant's  distance  from  A. 

Since  C  is  the  pole  of  B'A',  it 
follows  that  B'  is  also  a'  quad- 
rant's distance  from  C. 

.'.    B'  is  the  pole  of  AC,  §  563. 

Similarly,  prove  that  A'  is  the  pole  of  BC  and  C  is 
the  pole  of  A  B. 

.'.     AABC  is  the  polar  triangle  of  AA'B'C. 


Fig.  218 


POLYEDRAL  ANGLES 


315 


EXERCISES 

1.  Make  a  sketch  of  the  polar  triangle  of  a  birectangular 
triangle  and  show  that  it  is  also  birectangular. 

2.  Show  that  the  polar  triangle  of  a  given  trirectangular 
triangle  is  identical  with  the  given  triangle. 

338.  Theorem:  In  two  polar  spherical  triangles  each 
angle  of  the  one  is  the  supplement  of  that  side  of  the  other  of 
which  it  is  the  pole. 

a! 


Fig.  219 

Given  the  polar  triangles  A'B'C  and  ABC,  Fig.  219, 
having  the  sides  equal  to  a',  b',  c'  and  a,  b,  c  respectively. 
To  prove  that  A  +a'  =  180°,  B+b'  =  180°,  C+c'  =  180/ 
A'+a  =  180°,  B'+b  =  180°,  C'+c  =  180°. 

Proof:    Let  the  sides  of  /.A,  produced  if  necessary, 
intersect  the  side  B'C  in  points  D  and  E  respectively. 

Since  Z  A  is  measured  by  arc  DE,  §  333,  A=DE. 

Since  ITE  =  90°  =  DC',  B7E+DC/  =  180°. 

i^+(jD£+JE'C,)  =  180o.        Why? 

DE+  (B^E+E£)  =  180°.        Why  ? 

DE+lfc' =  180°.        Why? 

.*.    A+a'  =  180°.        Why? 

Similarly,  B+b'  =  180°,  C+c'  =  180°,  etc. 


316 


THIRD-YEAR  MATHEMATICS 


EXERCISES 

1.  Find  the  sides  of  the  polar  triangle  of  a  triangle  whose 
angles  are  75°,  85°,  and  88°  respectively. 

2.  The  angles  of  a  spherical  triangle  are  88°,  125°,  and  96° 
respectively.     Find  the  sides  of  the  polar  triangle. 

339.  Theorem:    The  sum  of  the  angles  of  a  spherical 
triangle  is  less  than  six  and  greater  than  two  right  angles. 


Given  the  spherical  triangle 
ABC,  Fig.  220. 

To  prove  that  A  +  B+C<5±0°; 
A  +  J5  +  O1800. 

Proof:  1.  Let  AA'B'C  be  the 
polar  triangle  of  AABC. 

Then  A +a'  =  180° 

B+b'  =  180° 
C+c'  =  180° 


Adding, 

A+B+C+a'+b'+c'  =  540° 

...     A+£+C  =  540°-(a'+6'+c') 
or  ^+J5+C<540° 

2.  A+B+C+a'+b'+c'  =  5±0° 

360°>a'+6'+c' 

Adding, 


(See  p.  307.) 


A+B+C+a'+b'+c'+ZQ0°>540o+a'+b'+c' 
.'.    A+B+C>1$0° 

340.  Spherical  excess.  The  amount  by  which  the 
sum  of  the  angles  of  a  spherical  triangle  exceeds  180°  is 
called  the  spherical  excess. 


POLYEDRAL  ANGLES 


317 


EXERCISES 

1.  Show  that  the  spherical  excess  of  a  triangle  is  equal  to 
Ao+B°+Co-180°. 

2.  The  angles  of  a  spherical  triangle  are  100°,  65°,  and  190°. 
Find  the  spherical  excess. 

Symmetry  and  Congruence 

341.  Congruent  polyedral  angles.  Two  polyedral 
angles  are  congruent  if  they  can  be  made  to  coincide. 

It  follows  that  the  face  angles  and  diedral  angles  of 
one  of  two  congruent  polyedral  angles  are  equal  respec- 
tively to  those  of  the  other.  However,  it  does  not  follow 
that  two  polyedral  angles  are  congruent  if  the  correspond- 
ing face  angles  and  diedral  angles  are  equal. 

For  example,  triedral  angles  O-ABC,  O'-A'B'C,  and 
0"-A"B"C",  Fig.  221,  have  the  corresponding  face 
angles  and  diedral  angles  equal. 


Fig.  221 


O-ABC  and  O'-A'B'C  are  congruent,  but  O-ABC 
and  0"-A"B"C"  are  not  congruent. 

342.  Symmetrical  polyedral  angles.  Two  polyedral 
angles  are  symmetrical  if  the  face  angles  and  diedral 
angles  of  one  are  equal  respectively  to  the  face  angles 
and  diedral  angles  of  the  other,  but  arranged  in  opposite 
order. 


318 


THIRD-YEAR  MATHEMATICS 


343.  Congruent  and  symmetrical  spherical  polygons. 

If  the  sides  and  angles  of  one  spherical  polygon  are  equal 
respectively  to  those  of  another,  the  polygons  are  con- 
gruent, provided  the  parts  are  arranged  in  the  same  order, 
and  symmetrical,  provided  the  parts  are  arranged  in 
opposite  order. 


Thus,  in  Fig.  222,  ABC^  AA'B'C  and  both  triangles 
are  symmetrical  to  AA"B"C". 

344.  Theorem:  If  two  triedral  angles  have  the  three 
face  angles  of  one  equal  respectively  to  the  three  face  angles 
of  the  other,  the  corresponding  diedral  angles  are  equal. 


Given  the  triedral  angles  O-ABC  and  O'-A'B'C, 
Fig.  223,  having  £AOB=£A'0'B',  ZBOC=  /.B'O'C \ 
ZCOA=  /.CO' A'. 

To  prove  that  diedral  angle  AO  —  diedral  angle  A'Or, 
BO  =  B'0'1  and  CO=CO'. 


POLYEDRAL  ANGLES  319 

Proof:  Lay  off  OA=OB  =  OC  =  0'A' =0'B'  =  0'C. 
Draw  AB,  BC,  CA,  A'B',  B'C,  and  C'A'. 
Prove   that    AAOB^AA'O'B';    ABOCz AB'O'C; 
ACOA  ^  AC'O'A'. 

Prove  that  AABC^  AA'B'C. 
Take  OD  =  0'D'. 

Draw  DE  and  DF  perpendicular  to  AO  and  in  faces 
A  OB  and  AOC  respectively. 

Similarly,  draw  D'W  and  D'F'. 
Prove  that        AEDA  £  E'D'A';  AFDA  sg  AF'D'A'. 
Prove  that         AEAF §2  AE'A'F'. 
Prove  that         AEDF  sg  AE'D'F'. 
.'.  ZEDF=ZE'D'F'. 
.'.  Diedral  /.AO  =  diedral  Z A '0',  having  equal 
plane  angles. 

Similarly,  diedral  Z  B0  =  diedral  AB'O'. 
diedral  ZC0  =  diedral  Z  CO'. 

EXERCISES 

Prove  the  following: 

1.  Two  triedral  angles  are  congruent  if  the  face  angles  of  one  are 
equal  respectively  to  the  face  angles  of  the  other,  arranged  in  the 
same  order. 

2.  Two  triedral  angles  are  symmetrical  if  the  face  angles  of  one 
are  equal  respectively  to  the  face  angles  of  the  other,  arranged  in  the 
reverse  order. 

3.  If  two  spherical  triangles  on  the  same  sphere  or  on  equal  spheres 
have  three  sides  of  one  equal  respectively  to  three  sides  of  the  other — • 

1.  They  are  congruent  if  the  equal  parts  are  arranged  in  the 
same  order. 

2.  They  are  symmetrical  if  the  equal  parts  are  arranged  in  the 
reverse  order. 

Show  that  the  triangles  being  mutually  equilateral  are  also  mutu- 
ally equiangular,  and  therefore  either  congruent  or  symmetrical. 


320 


THIRD-YEAR  MATHEMATICS 


4.  Two  spherical  triangles  on  the  same  or  equal  spheres  are 
congruent — 

1.  //  two  sides  and  the  included  angle  of  one  are  equal  respec- 
tively to  two  sides  and  the  included  angle  of  the  other,  arranged  in  the 
same  order. 

2.  7/  two  angles  and  the  included  side  of  one  are  equal  respec- 
tively to  two  angles  and  the  included  side  of  the  other,  arranged  in 
the  same  order. 

Show  that  AABC,  Fig.  224,  can  be  maMe  to  coincide  with 

AA'B'C. 

a 

C 


>0 


+0' 


Fig.  224 


5.  Two  spherical  triangles  on  the  same  or  equal  spheres  are 
symmetrical — 

1 .  7/  two  sides  and  the  included  angle  of  one  are  equal  respec- 
tively to  two  sides  and  the  included  angle  of  the  other,  arranged 
in  the  reverse  order. 

2.  If  two  angles  and  the  included  side  of  one  are  equal  respec- 
tively to  two  angles  and  the  included  side  of  the  other,  arranged  in 
the  reverse  order. 


Let  AABC  and  A*B'C,  Fig.  225,  be  the  given  triangles. 
Draw  AA"B"C"  symmetrical  to  AA'B'C 


POLYEDRAL  ANGLES 


321 


Then  A  A'B'C  and  A"B"C"  are  mutually  equilateral  and 
mutually  equiangular. 

Prove  that  AABC&  AA"B"C". 

:.  AABC  is  symmetrical  to  AA'B'C. 

6.  State  and  prove  theorems  on  triedral  angles  correspond- 
ing to  the  theorems  in  exercises  3,  4,  and  5. 

7.  2/  two  spherical  triangles  on  the  same  or  equal  spheres  are 
mutually  equiangular,  they  are  mutually  equilateral  and  congruent 
if  the  equal  parts  are  arranged  in  the  same  order.  If  the  equal 
parts  are  arranged  in  the  reverse  order,  they  are  symmetrical. 


Fig.  226 


Given  the  spherical  triangles  ABC  and  A'B'C,  Fig.  226,  hav- 
ing ZA  =  ZA',  ZB=ZB',  ZC=ZC. 

To  prove  that  a  =  a',  b  =  b',  c  =  c',  and  that  AABC  and  A'B'C 
are  either  congruent  or  symmetrical. 

Proof:   Construct  the  polar  triangles  DEF  and  D'E'F'. 

Show  that  ADEF  and  D'E'F'  are  mutually  equilateral,  §  338. 

Show  that  A  DEF  and  D'E'F'  are  mutually  equiangular,  exercise  3 . 

Show  that  AABC  and  A'B'C  are  mutu- 
ally equilateral,  §338.  ^C 

.'.  ABC  and  A'B'C  are  either  congruent 
or  symmetrical,  §  343. 


345.  Theorem:  The  angles  opposite 
the  equal  sides  of  an  isosceles  spherical 
triangle  are  equal.  v 

Draw  the  arc  of  a  great  circle,  CD,  b 

Fig.  227,  bisecting  the  side  AB.  Fig.  227 


;*0 


322 


THIRD-YEAR  MATHEMATICS 


Then  A  A  DC  and  BDC  are  mutually  equilateral  and 
therefore  symmetrical, 

.*.  '  ZA=ZB 


EXERCISES 

1.  State  and  prove  the  theorem  on  triedral  angles  cor- 
responding to  §  345. 

2.  Two  symmetrical  isosceles  triangles  are  congruent.    Prove. 

346.  Theorem:  If  two  angles 
of  a  spherical  triangle  are  equal, 
the  sides  opposite  are  equal. 

Construct  the  polar  triangle 
of  AABC,  Fig.  228. 

Since  ZA=  ZB, 

it  follows  that  B'C'  =  A'C',  §  338. 
.*.     ZB'=Z  A',  §345. 
.*.    AC  =  BC,  §338. 


Fig.  228 


347.  Theorem:  If  two  angles  of  a  spherical  triangle  are 
unequal,  the  sides  opposite  are  unequal  and  the  greater  side 
lies  opposite  the  greater  angle. 

Draw  AD,  Fig.  229,  an  arc  of 
a  great  circle  making 

ZDAB=ZDBA. 


Then 


or 


AC<CD+DA, 

AC<CD+DB. 

.'.    AC<CB. 


Fig.  229 


348.  Theorem:  //  two  sides  of  a  spherical  triangle  are 
unequal,  the  angles  opposite  are  unequal  and  the  greater 
angle  lies  opposite  the  greater  side. 

Prove  by  the  indirect  method. 


POLYEDRAL  ANGLES 


323 


349.  Theorem:  The  diameters  of  a 
through  the  vertices  of  a  given 
spherical  triangle  meet  the 
surface  of  the  sphere  in  points 
which  are  the  vertices  of  a 
triangle  symmetrical  to  the  given 
triangle. 

Prove  that  A  ABC  and  A'B'C, 
Fig.  230,  are  mutually  equilateral 
and  therefore  mutually  equi- 
angular. Fig.  230 


lere    drawn 


Area  of  a  Spherical  Triangle 

350.  Theorem:     Two  symmetrical  spherical  triangles 
are  equal. 

Given  the  symmetrical 
spherical  triangles  ABC  and 
A'B'C,  Fig.  231. 

To  prove  that 

AABC  =  AA'B'C. 

Proof:  Let  the  two  triangles 
be  placed  so  that  A  and  Ar,  B 
and  B',  C  and  Cf  are  opposite  •    Fig.  231 

endpoints  of  diameters.   Let  P  be 

the  pole  of  the  small  circle  determined  by  points  A}  B, 
and  C,  and  let  P'  be  the  other  endpoint  of  the  diameter 
through  P. 

Show  that   AAPB  and  A'P'B'  are  isosceles,  sym- 
metrical, and  therefore  congruent. 

Similarly,  ACPA  g  AC'P'A' 

and  ABPC&AB'P'C. 


Adding, 


AABC  =  AA'B'C. 


324  THIRD-YEAR  MATHEMATICS 

351.  Lune.  Two  great  semicircles,  ACB  and  ADB, 
Fig.  232,  form  a  lune.  The 
spherical  angle  CAD  is  the 
angle  of  the  lune.  The  portion  of 
the  surface  of  the  sphere  included 
between  the  semicircles  is  the 
surface  of  the  lune,  and  its  area  is 
the  area  of  the  lune. 

352.  Spherical  degree.    The 

area  of  a  spherical  triangle  having 

two  sides  equal  to  quadrants  and 

the  included  spherical  angle  equal 

to  1°  is  used  as  the  unit  of  measure  of  areas  of  spherical 

polygons.     It  is  called  a  spherical  degree. 

EXERCISES 

1.  Show  that  a  trirectangular  triangle  contains  90  spherical 
degrees. 

2.  Show  that  the  surface  of  a  sphere  contains  720  spherical 
degrees. 

3.  The  number  of  spherical  degrees  in  the  surface  of  a  lune  is 
twice  the  number  of  degrees  in  its  angle,  i.e., 

L  =  2(A), 

where  A  is  the  number  of  degrees  in  the  angle  of  the  lune  and 
L  the  number  of  spherical  degrees  in  the  surface  of  the  lune. 

4.  Show  that  the  area  of  a  spherical  degree,  in  units  of  plane 
surface,  is         . 

5.  Show  that  the  area,  S,  of  a  lune,  in  units  of  plane  surface, 

— - ,  where  A  is  the  number  of 
90    ' 

lune  and  R  the  radius  of  the  sphere. 


is  *"*       ,  where  A  is  the  number  of  degrees  in  the  angle  of  the 


POLYEDRAL  ANGLES  325 

353.,  Theorem:    The  area,  in  spherical  degrees,  of  a 
spherical  triangle  is  equal  to  the  spherical  excess. 

Given  the  spherical  triangle  ^^i^^^ 

ABC,  Fig.  233.  jT                kj. 

To  prove  that  the  area,  S,  M                        I  k 

of    AABC   is  given   by  the  rm                            m 

formula  |<^           —^       m 

£=(A+£+C-180)  V  ^ 

spherical  degrees,  C^C  >^ 

A,  B,  and  C  being  the  number  Fig.  233 

of  degrees  in  the  angles   of 

AABC   and    S   the    number    of    spherical    degrees    in 

AABC. 

Proof:  Extend  the  sides  of  AABC. 
The  angles  of  AABC  are  the  angles  of  lunes  C'ACBC, 
B'CBAB',  and  A'BACA'  respectively. 

Lune       CACBC  =  AABC + AABC  =  2(C),  exercise 
3,  §  352. 

Lune  B'CBABf  =  AABC+  ACB'A  =  2(B) 
Lune  A'BACA' =  AABC+ACA'B  =  2(A) 

S(AABC)  +  AABC'+ACB'A  +  ACA'B  =  2(A+B+C) 

or  (2  AABC)  +  AABC  +  AABC + AC  B' A  +  AC  A' B 

=  2(A+B+C) 

Show  that  ACA'B  =  ACAB' 

.\2(AABC)  +  AABC+AABC+ACB'A  +  AC'AB' 

=  2(A+B+C) 

.-.  2(AA£C)+hemisphere  =  2(A+J5+C) 

.'.  2(AABC)+%(720)=2(A+B+C) 

.'.  AABC  =  A+B+C-i(720) 

or  S=A+B+C-1S0. 


326 


THIRD-YEAR  MATHEMATICS 


354.  Let  ABODE,  Fig.  234,  be  a  spherical  polygon  of  n 
sides. 

Divide  ABODE  into  n-2 
spherical  triangles  by  drawing 
arcs  of  great  circles,  as  AC  and 
AD. 

Denoting  by  Th  T2,  Th  etc., 
the  areas  of  ABAC,  CAD,  DAE, 
etc.,  and  by  sh  s2,  S3,  etc.,  the 
sums  of  the  angles  in  triangles 
BAC,  CAD,  DAE,  etc.,  it  follows 
that 

Ti  =  si-180, 
T2  =  s2-180, 
T3  =  s3-180,  etc. 


Fig.  234 


Adding, 


P  =  [si+S2+s3+etc.-(rc-2)180], 

or  P  =  s-(n-2)180, 

where  P  denotes  the  number  of  spherical  degrees  in  the 

polygon  and  s  the  number  of  degrees  in  the  sum  of  the 

angles. 

This  may  be  stated  as  a  theorem  as  follows : 

The  area  of  a  spherical  polygon  is  equal  to  its  spherical 

excess. 

EXERCISES 

1.  The  angles  of  a  spherical  triangle  are  90°,  90°,  and  79°. 
Find  the  area  in  spherical  degrees. 

2.  Find  the  area  of  a  spherical  degree  on  the  spheres  having 
radius  equal  to  3  in. ;   14  in. ;  a  inches. 

3.  Find  the  area  of  a  spherical  polygon  whose  angles  are  70°, 
105°,  145°,  125°,  150°. 

4.  Find  the  area  of  a  spherical  triangle  whose  angles  are 
85°,  120°,  and  95°  on  a  sphere  whose  radius  is  6  inches. 


POLYEDRAL  ANGLES  327 

5.  The  area  of  a  spherical  triangle  is  100  sq.  in.  and  its 
angles  are  100°,  64°,  200°.  What  is  the  radius  of  the  sphere  on 
which  the  triangle  lies  ?     (Board.) 

6.  Prove  that  the  area  of  a  spherical  triangle  is  proportional 
to  its  spherical  excess. 

Three  complete  great  circles  drawn  on  a  sphere  whose 
radius  is  10  in.  divide  the  surface  of  the  sphere  into  eight 
spherical  triangles;  the  angles  of  one  of  these  triangles  are  40°, 
80°,  and  120°.  Find  the  area  in  square  inches  of  each  of  the 
eight  triangles.     (Harvard.) 

7.  On  a  sphere  of  radius  2  ft.  the  area  of  a  certain  triangle 
is  2  square  yards.  What  is  the  perimeter  of  the  polar  triangle  ? 
(Harvard.) 

Summary 

355.  The  chapter  has  taught  the  meaning  of  the 
following  terms : 

polyedral  angle,  triedral  angle  polar  spherical  triangles 

spherical  polygon  spherical  excess 

spherical  triangle  congruent  and  symmetrical 
similar  polyedrons  figures 

spherical  angle  lune 

right,    birectangular,    and    tri-  spherical  degree 
rectangular  spherical  triangles 

356.  The  following  theorems  have  been  studied: 

1.  The  sum  of  two  face  angles  of  a  triedral  angle  is 
greater  than  the  third. 

2.  The  sum  of  two  sides  of  a  spherical  triangle  is  greater 
than  the  third  side. 

3.  The  shortest  line  that  can  be  drawn  between  two  given 
points  on  the  surface  of  a  sphere  is  the  minor  arc  of  the  great 
circle  which  passes  through  the  two  points. 


328  THIRD-YEAR  MATHEMATICS 

4.  The  sum  of  the  face  angles  of  a  convex  polyedral  angle 
is  less  than  four  right  angles. 

5.  The  sum  of  the  sides  of  a  convex  spherical  polygon 
is  less  thqn  360°. 

6.  There  cannot  be  more  than  five  kinds  of  regular 
polyedrons. 

7.  Two  tetraedrons  having  a  triedral  angle  of  one  equal 
to  a  triedral  angle  of  the  other  are  to  each  other  as  the  products 
of  the  edges  including  the  equal  triedral  angles. 

8.  Two  similar  polyedrons  are  to  each  other  as  the  cubes 
of  two  corresponding  edges. 

9.  A  spherical  angle  is  measured  by  the  arc  of  a  great 
circle  having  the  vertex  as  a  pole^and  included  between  the 
sides  produced  if  necessary. 

10.  A  spherical  angle  is  equal  to  the  diedral  angle  formed 
by  the  planes  of  the  sides. 

11.  If  a  spherical  triangle  is  the  polar  triangle  of  another, 
then  the  second  is  the  polar  triangle  of  the  first. 

12.  In  two  polar  spherical  triangles  each  angle  of  the  one 
is  the  supplement  of  that  side  of  the  other  of  which  it  is  the 
pole. 

13.  The  sum  of  the  angles  of  a  spherical  triangle  is  less 
than  six  and  greater  than  two  right  angles. 

14.  If  two  triedral  angles  have  the  corresponding  face 
angles  equal,  the  corresponding  diedral  angles  are  equal. 

15.  (1)  If  the  face  angles  of  one  triedral  angle  are  equal 
respectively  to  the  face  angles  of  another; 


POLYEDRAL  ANGLES  329 

(2)  If  two  face  angles  and  the  included  diedral  angle  are 
equal  respectively  to  the  corresponding  parts  of  the  other; 

(3)  If  two  diedral  angles  and  the  included  face  angle  are 
equal  respectively  to  the  corresponding  parts  of  the  other; 

(4)  If  the  diedral  angles  of  one  are  equal  respectively  to 
the  corresponding  parts  of  the  other; 

The  triedral  angles  are  congruent  if  the  parts  are  arranged 
in  the  same  order,  and  symmetrical  if  they  are  arranged  in 
the  reverse  order.      v 

16.  Two  spherical  triangles  are  congruent,  or  symmetri- 
cal, if  they  have  the  following  corresponding  parts  equal: 

(1)  Three  sides;  (2)  three  angles;  (3)  two  sides  and  the 
included  angle;   (4)  two  angles  and  the  included  side. 

17.  If  two  spherical  triangles  on  the  same  or  equal  spheres 
are  mutually  equiangular,  they  are  mutually  equilateral,  and 
conversely. 

18.  The  base  angles  of  an  isosceles  spherical  triangle  are 
equal,  and  conversely. 

19.  If  two  angles  of  a  spherical  triangle  are  unequal,  the 
sides  opposite  are  unequal,  and  conversely. 

20.  The  diameters  of  a  sphere  drawn  through  the  vertices 
of  a  spherical  triangle  meet  the  surface  of  the  sphere  in  points 
which  are  the  vertices  of  a  triangle  symmetrical  to  the  given 
triangle. 

21.  Two  symmetrical  spherical  triangles  are  equal. 
357.  The  following  constructions  were  taught: 

1.  To  construct  a  sphere  passing  through  four  given 
points  not  all  in  the  same  plane. 

2.  To  inscribe  a  sphere  in  a  given  tetraedron. 

3.  To  determine  the  diameter  of  a  given  material  sphere. 


330  THIRD-YEAR  MATHEMATICS 

358.  The  following  formulas  have  been  proved: 

1.  The  area  of  a  lune, 

L  =  2(A)  spherical  degrees, 

where  A  is  the  number  of  degrees  in  the  angle  of  the  lune. 

2.  The  area  of  a  spherical  triangle, 

T  =  (A+B+C -180)  spherical  degrees, 

where  A,  B,  and  C  are  the  number  of  degrees  in  the  angles 
of  the  triangle. 

This  formula  may  be  stated, 

T=E,  the  spherical  excess. 

3.  The  area  of  a  spherical  polygon, 

P  =  (s  —  (n — 2)  180)  spherical  degrees, 

where  s  is  the  number  of  degrees  in  the  sum  of  the  angles 
of  the  polygon. 

This  formula  may  be  stated, 

P=E,  the  spherical  excess. 


CHAPTER  XV 

SUMMARY  OF  THE  ASSUMPTIONS  AND  THEOREMS 

OF  GEOMETRY  GIVEN  IN  THE  COURSES  OF  THE 

FIRST  AND  SECOND  YEARS 

359.  For  the  convenience  of  the  student  a  complete 
list  of  the  assumptions  and  theorems  studied  in  the  first 
two  courses  is  given  below.  References  in  the  foregoing 
chapters  are  made  to  this  list  to  save  the  student  the  time 
of  looking  them  up  in  other  textbooks.  The  numbers  in 
the  parentheses  (  )  refer  to  the  sections  in  First-Year 
Mathematics,  those  in  brackets  [  ]  to  the  sections  in  Second- 
Year  Mathematics  in  which  these  statements  were  given 
for  the  first  time. 

Preliminary  Assumptions 

360.  Through  two  points  one  and  only  one  straight 
line  can  be  drawn.     (20) 

361.  A  straight  line  two  of  whose  points  lie  in  a  plane 
lies  entirely  in  the  plane.     (204) 

362.  The  shortest  distance  between  two  points  is  the 
straight  line-segment  joining  the  points.     (21) 

363.  Two  straight  lines  intersect  in  one  and  only  one 
point.     (25) 

364.  A  line-segment,  or  an  angle,  is  equal  to  the  sum 
of  all  its  parts.     (33)      %<w-    .'-/■- 

365.  A  segment,  or  an  angle,  is  greater  than  any  of  its 
parts,  if  only  positive  magnitudes  are  considered.     (34) 

366.  If  the  same  number  is  added  to  equal  numbers, 
the  sums  are  equal.     (35) 

331  • 


332  THIRD-YEAR  MATHEMATICS 

367.  If  equals  are  added  to  equals,  the  sums  are  equal.. 
(36) 

368.  If  the  same  number  or  equal  numbers  be  sub- 
tracted from  equal  numbers,  the  differences  are  equal. 
(41) 

369.  The  sums  obtained  by  adding  unequals  to  equals 
are  unequal  in  the  same  order  as  are  the  unequal  addends. 

(42) 

370.  The  sums  obtained  by  adding  unequals  to  un- 
equals in  the  same  order  are  unequal  in  the  same  order. 
(43) 

371.  The  differences  obtained  by  subtracting  unequals 
from  equals  are  unequal  in  the  order  opposite  to  that  of 
the  subtrahends.     (44) 

372.  If  equals  be  divided  by  equal  numbers  (excluding 
division  by  0),  the  quotients  are  equal.     (78) 

373.  If  equals  be  multiplied  by  the  same  number  or 
equal  numbers,  the  products  are  equal.     (80) 

Angles 

374.  All  right  angles  are  equal.     (118) 

\J  375.  Equal  central  angles  in  the  same  or  equal  circles 

intercept  equal  arcs.     (124) 

376.  In  the  same  or  equal  circles  equal  arcs  are  inter- 
cepted by  equal  central  angles.     (125) 

377.  A  central  angle  is  measured  by  the  intercepted 
arc.     (126) 

378.  If  two  angles  have  their  sides  parallel  respectively 
they  are  equal  or  supplementary.     (197) 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     333 

379.  If  the  sum  of  two  adjacent  angles  is  a  straight 
angle,  the  exterior  sides  are  in  the  same  straight  line. 
(177) 

380.  The  sum  of  all  the  adjacent  angles  about  a  point, 
on  one  side  of  a  straight  line,  is  a  straight  angle.     (179) 

381.  The  sum  of  all  the  angles  at  a  point  just  covering 
the  angular  space  about  the  point  is  a  perigon.     (180) 

382.  If  two  lines  intersect,  the  opposite  angles  are 
equal.     (183) 

Angles  of  a  Triangle  and  Polygon 

383.  The  sum  of  the  angles  of  a  triangle  is  180°.  (112), 
(198) 

384.  The  sum  of  the  exterior  angles  of  a  triangle,  tak- 
ing one  at  each  vertex,  is  360°.     (115) 

385.  An  exterior  angle  of  a  triangle  is  equal  to  the  sum 
of  the  two  remote  interior  angles.     (118),  (199) 

386.  If  the  angles  of  one  triangle  are  respectively  equal 
to  the  angles  of  another,  the  third  angles  are  equal.     (281) 

387.  The  base  angles  of  an  isosceles  triangle  are  equal. 
(280) 

388.  An  equilateral  triangle  is  equiangular.     (281) 

389.  If  two  angles  of  a  triangle  are  equal,  the  triangle 
is  isosceles.     (281) 

390.  The  acute  angles  of  a  right  triangle  are  comple- 
mentary angles.     (184) 

391.  In  a  right  triangle  whose  acute  angles  are  30°  and 
60°  the  side  opposite  the  90°-angle  is  twice  as  long  as  the 
side  opposite  the  30°-angle.     (185) 


334  THIRD-YEAR  MATHEMATICS 


J 


392.  The  sum  of  the  interior  angles  of  a  polygon  is 
(n  —  2)  straight  angles.     [88] 

^  393.  The  sum  of  the  exterior  angles  of  a  polygon, 

taking  one  at  each  vertex,  is  360°.     [89] 

Perpendicular  Lines 

394.  The  shortest  distance  from  a  point  to  a  line  is 
the  perpendicular  from  the  point  to  the  line.     (285) 

395.  At  a  given  point  in  a  given  line  one  and  only  one 
perpendicular  can  be  drawn  to  the  line.     (176) 

396.  From  a  given  point  without  a  straight  line  one 
perpendicular  can  be  drawn  to  the  line,  and  only  one. 
[296] 

397.  All  points  on  the  perpendicular  bisector  of  a 
line-segment  are  equidistant  from  the  endpoints  of  the 
segment.     (281) 

398.  If  a  point  is  equidistant  from  the  endpoints  of  a 
line-segment,  it  is  on  the  perpendicular  bisector  of  the 
segment.     (283) 

399.  If  each  of  two  points  on  a  given  line  is  equally 
distant  from  two  given  points,  the  given  line  is  the  per- 
pendicular bisector  of  the  segment  joining  the  given 
points.     [83] 

Parallel  Lines 

400.  Parallel  lines  are  everywhere  equally  distant. 

(192) 

401.  One  and  only  one  parallel  can  be  drawn  to  a  line 
from  a  point  outside  the  line.     (194) 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     335 

402.  If  two  lines  are  cut  by  a  transversal  making  the 
corresponding  angles  equal,  the  lines  are  parallel.     (195) 

403.  Two  lines  perpendicular  to  the  same  line  are 
parallel.     [110] 

404.  Two  lines  are  parallel  if  two  alternate  interior 
angles  formed  with  a  transversal  are  equal.     [112] 

405.  Two  lines  are  parallel  if  the  interior  angles  on  the 
same  side  formed  with  a  transversal  are  supplementary. 
[116] 

406.  Two  lines  parallel  to  the  same  line  are  parallel 
to  each  other.     (195) 

407.  If  two  parallel  lines  are  cut  by  a  transversal,  the 
corresponding  angles  are  equal;  the  alternate  interior 
angles  are  equal;  the  interior  angles  on  the  same  side  are 
supplementary.     (196) 

408.  A  line  perpendicular  to  one  of  two  parallel  lines 
is  perpendicular  to  the  other.     [116] 

409.  A  line  bisecting  a  side  of  a  triangle  and  parallel 
to  a  second  side  bisects  the  third  side.     [159] 

410.  If  three  or  more  parallel  lines  intercept  equal 
segments  on  one  transversal,  they  intercept  equal  segments 
on  every  transversal.     [160] 

411.  The  line  joining  the  midpoints  of  two  sides  of  a 
triangle  is  parallel  to  the  third  side.     [168] 

Congruent  Triangles 

412.  Two  triangles  are  congruent  if  two  sides  and  the 
included  angle  of  one  are  equal  respectively  to  two  sides 
and  the  included  angle  of  the  other,     (s.a.s.)     (274) 


336  THIRD-YEAR  MATHEMATICS 

413.  Two  triangles  are  congruent  if  two  angles  and 
the  side  included  between  their  vertices  in  one  are  equal 
respectively  to  the   corresponding  parts  in  the  other. 

(a.s.a.)     (275) 

414.  If  three  sides  of  one  triangle  are  equal,  respec- 
tively, to  the  three  sides  of  another  triangle,  the  triangles 
are  congruent,     (s.s.s.)     (283) 

415.  Two  right  triangles  are  congruent  if  the  hypot- 
enuse and  one  side  of  one  are  equal  respectively  to  the 
hypotenuse  and  a  side  of  the  other.     (285) 

Quadrilaterals 

416.  If  a  quadrilateral  is  a  parallelogram — 

1.  A  diagonal  divides  it  into  congruent  triangles; 

2.  The  opposite  sides  are  equal; 

3.  The  opposite  angles  are  equal; 

4.  The  consecutive  angles  are  supplementary; 

5.  The  diagonals  bisect  each  other.     [118-122] 

417.  A  quadrilateral  is  a  parallelogram  if — 

1.  The  opposite  sides  are  parallel; 

2.  The  opposite  sides  are  equal; 

3.  One  pair  of  opposite  sides   are  equal   and 
parallel; 

4.  The  opposite  angles  are  equal; 

5.  The  diagonals  bisect  each  other.     [123-127] 

Similar  Figures 

418.  A  line  parallel  to  one  side  of  a  triangle  forms  with 
the  other  two  sides  a  triangle  similar  to  the  given  triangle. 
[214] 


ASSUMPTIONS  f  ND  THEOREMS  OF  GEOMETRY     337 

419.  Two  triangles  are  similar  if  two  angles  of  one  are 
respectively  equal  to  two  angles  of  the  other.     [217] 

420.  Two  triangles  are  similar  if  the  ratio  of  two  sides 
of  one  equals  the  ratio  of  two  sides  of  the  other  and  the 
angles  included  between  these  sides  are  equal.     [218] 

421.  Two  triangles  are  similar  if  the  corresponding 
sides  are  in  proportion.     [219] 

422.  The  perimeters  of  similar  polygons  are  to  each 
other  as  any  two  homologous  sides.     [219] 

423.  Similar  polygons  may  be  divided  by  homologous 
diagonals  into  triangles  similar  to  each  other  and  similarly 
placed.     [220] 

424.  The  perpendicular  to  the  hypotenuse  from  the 
vertex  of  the  right  angle  divides  a  right  triangle  into  parts 
similar  to  each  other  and  to  the  given  triangle.     [224] 


Relations  between  the  Sides  of  a  Triangle 

425.  In  a  right  triangle  the  square,  of  the  hypotenuse 

is  equal  to  the  sum  of  the  squares  of  the  sides  of  the  right    »   ^ 
angle.     [233],  algebraic  proof;  [462],  geometric  proof. 

426.  In  a  triangle  the  square  on  the  side  opposite  an   \f 
acute  angle  is  equal  to  the  sum  of  the  squares  of  the  other 

two  sides  diminished  by  two  times  the  product  of  one  of 
these  two  sides  and  the  projection  of  the  other  upon  it. 

[240] 

427.  In  a  triangle  the  square  on  the  side  opposite  the      ^ 
obtuse  angle  is  equal  to  the  sum  of  the  squares  on  the  other 
two  sides,  increased  by  two  times  the  product  of  one  of 
them  and  the  projection  of  the  other  upon  it.     [241] 


338  THIRD-YEAR  MATHEMATICS 

428.  In  a  triangle  the  sum  of  the  squares  of  two  sides 
is  equal  to  twice  the  square  of  one-half  of  the  third  side 
increased  by  twice  the  square  of  the  median  to  the  third 
side. 

Proportional  Line-Segments 

429.  The  perimeters  of  similar  polygons  are  in  the 
same  ratio  as  any  two  corresponding  sides.     [219] 

430.  If  two  chords  of  a  circle  intersect,  the  product  of 
the  segments  of  one  is  equal  to  the  product  of  the  segnients 
of  the  other.     [314] 

431.  If  from  a  point  without  a  circle  a  tangent  and 
secant  be  drawn,  the  tangent  is  a  mean  proportional 
between  the  entire  secant  to  the  concave  arc  and  the 
external  segment.     [315] 

432.  If  from  a  point  without  a  circle  two  secants  be 
drawn  to  the  concave  arc,  the  product  of  one  secant  and 
its  external  segment  is  equal  to  the  product  of  the  other 
secant  aA  q  its  external  segment.     [317] 

433.  In  a  right  triangle  the  perpendicular  from  the 
vertex  of  the  right  angle  to  the  hypotenuse  is  the  mean  pro- 
portional between  the  segments  of  the  hypotenuse.     [230] 

434.  In  a  right  triangle  either  side  of  the  right  angle 
is  the  mean  proportional  between  its  projection  upon  the 
hypotenuse  and  the  entire  hypotenuse.     [232] 

435.  A  perpendicular  to  a  diameter  of  a  circle  at  any 
point,  extended  to  the  circle,  is  the  mean  proportional 
between  the  segments  of  the  diameter.     [231] 

436.  If  two  parallels  cut  two  intersecting  transversals, 
the  segments  intercepted  on  one  transversal  are  propor- 
tional to  the  corresponding  segments  on  the  other.     [163] 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     339 

437.  If  a  number  of  parallels  cut  two  transversals,  the 
segments  intercepted  on  one  transversal  are  proportional 
to  the  corresponding  segments  on  the  other.     [167] 

438.  Two  lines  that  cut  two  given  intersecting  lines 
and  make  the  corresponding  segments  of  the  given  lines 
proportional  are  parallel.     [167] 

439.  The  bisector  of  an  angle  of  a  triangle  divides  the 
opposite  side  into  segments  proportional  to  the  adjacent 
sides.     [170] 

Circles 

440.  A  point  is  within,  upon,  or  without  a  circle, 
according  as  its  distance  from  the  center  is  less  than,  equal 
to,  or  greater  than  the  radius.     [274] 

441.  Circles  having  equal  radii  are  equal,  and  equal 
circles  have  equal  radii.     [275] 

442.  A  diameter  divides  a  circle  into  equal  parts. 

[276] 

443.  The  radius  drawn  to  the  point  of  contact  of  a 
tangent  is  perpendicular  to  the  tangent.     (308) 

444.  A  line  perpendicular  to  a  radius  at  the  outer 
endpoint  is  tangent  to  the  circle.     (309) 

445.  A  circle  can  be  drawn  through  three  points  not 
in  a  straight  line.     (312) 

446.  In  the  same  or  equal  circles  equal  central  angles 
intercept  equal  arcs  and  equal  arcs  are  intercepted  by   *^ 
equal  central  angles.     [281] 

447.  In  the  same  or  equal  circles  equal  arcs  are  sub-      i  / 
tended  by  equal  chords,    and,  conversely,  equal  chords 
subtend  equal  arcs.     [283] 


340  THIRD-YEAR  MATHEMATICS 

448.  If  two  tangents  to  the  same  circle  intersect,  the 
distances  from  the  point-  of  intersection  to  the  points  of 
contact  are  equal.     [315] 

449.  If  any  two  of  the  following  five  conditions  are 
taken  as  hypothesis,  the  remaining  three  are  true : 

1.  A  line  passes  through  the  center. 

2.  A  line  is  perpendicular  to  a  chord. 

3.  A  chord  is  bisected  by  a  line. 

4.  A  minor  arc  is  bisected. 

5.  A  major  arc  is  bisected.     [284] 

450.  In  the  same  or  equal  circles  equal  chords  are 
equally  distant  from  the  center,  and,  conversely,  chords 
equally  distant  from  the  center  are  equal.     [285] 

451.  The  arcs  included  between  two  parallel  secants 
are  equal;  and,  conversely,  if  two  secants  include  equal 
arcs  and  do  not  intersect  within  the  circle,  they  are 
parallel.     [286] 

452.  The  line  joining  the  centers  of  two  intersecting 
circles  bisects  the  common  chord  perpendicularly.     [287] 

453.  If  two  circles  are  tangent  to  each  other,  the 
centers  and  the  point  of  tangency  lie  in  a  straight  line. 
[289] 

Measurement  of  Angles  by  Arcs 

454.  A  central  angle  is  measured  by  the  intercepted 
arc.     [297] 

455.  In  the  same  or  equal  circles  two  central  angles 
have  the  same  ratio  as  the  intercepted  arcs.     [297] 

456.  An  inscribed  angle  is  measured  by  one-half  the 
arc  intercepted  by  the  sides.     [298] 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     341 

457.  An  angle  formed  by  a  tangent  and  a  chord  passing 
through  the  point  of  contact  is  measured  by  one-half  of 
the  intercepted  arc.     [300] 

458.  If  two  chords  intersect  within  a  circle,  either 
angle  formed  is  measured  by  one-half  the  sum  of  the  inter- 
cepted arcs.     [304] 

459.  If  two  secants  meet  outside  of  a  circle,  the  angle 
formed  is  measured  by  one-half  the  difference  of  the  inter- 
cepted arcs.     [305] 

460.  The  angle  formed  by  a  tangent  and  a  secant 
meeting  outside  of  a  circle  is  measured  by  one-half  the 
difference  of  the  intercepted  arcs.     [306] 

461.  The  angle  formed  by  two  tangents  to  a  circle 
is  equal  to  one-half  the  difference  of  the  intercepted  arcs. 
[307] 

Regular  Polygons  and  the  Circle 

462.  If  a  circle  is  divided  into  equal  arcs,  the  chords 
subtending  these  arcs  form  a  regular  inscribed  polygon. 
[435] 

463.  If  the  midpoints  of  the  arcs  subtended  by  the 
sides  of  a  regular  inscribed  polygon  of  n  sides  are  joined 
to  the  adjacent  vertices  of  the  polygon,  a  regular  inscribed 
polygon  of  2n  sides  is  formed.     [436] 

464.  If  a  circle  is  divided  into  equal  arcs,  the  tangents 
drawn  at  the  points  of  division  form  a  regular  circum- 
scribed polygon.     [437] 

465.  If  tangents  are  drawn  to  a  circle  at  the  midpoints 
of  the  arcs  terminated  by  consecutive  points  of  contact 
of  the  sides  of  a  regular  circumscribed  polygon,  a  regular 
circumscribed  polygon  is  formed  having  double  the 
number  of  sides.     [438] 


342  THIRD-YEAR  MATHEMATICS 

466.  If  at  the  midpoints  of  the  arcs  subtended  by  the 
sides  of  a  given  regular  inscribed  polygon  tangents  are 
drawn  to  the  circle,  they  are  parallel  to  the  sides  of  the 
given  polygon  and  form  a  regular  circumscribed  polygon. 
[446] 

467.  A  circle  may  be  circumscribed  about  any  'given 
regular  polygon.     [445] 

468.  A  circle  may  be  inscribed  in  any  given  regular 
polygon.     [446] 

469.  The  perimeter  of  a  regular  inscribed  2n-side  is 

greater  than  the  perimeter  of  the  regular  n-side  inscribed 

in  the  same  circle.     [447] 

«"? 

470.  The  perimeter  of  a  regular  circumscribed  2n-side 
is  less  than  the  perimeter  of  the  regular  n-side  circum- 
scribed about  the  same  circle.     [448] 

471.  The  length  of  a  circle  is  expressed  by  the  formula 

C  =  ird,  orC  =  2irr.    [450] 

Inequalities 

472.  The  diameter  oi  !a  circle  is  larger  than  any  other 
chord  of  the  circle.     [339]  <* 

473.  An  exterior  angle  of  a  triangle  is  greater  than 
either  of  the  remote  interior  angles.     [339] 

474.  If  two  sides  of  a  triangle  are  unequal,  the  angles 
opposite  to  them  are  unequal,  the  greater  angle  lying 
opposite  the  greater  side.     (281) 

475.  If  two  angles  of  a  triangle  are  unequal,  the  sides 
opposite  to  them  are  unequal,  the  greater  side  lying 
opposite  the  greater  angle.     (281) 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     343 

476.  If  two  oblique  line-segments  drawn  to  a  line  from 
a  point  on  a  perpendicular  to  the  line  have  unequal  pro- 
jections, the  oblique  line-segments  are  unequal.     [340] 

477.  Two  unequal  oblique  line-segments  drawn  to  a 
line  from  a  point  on  a  perpendicular  to  the  line  have 
unequal  projections.     [341] 

478.  If  from  a  point  inside  a  triangle  line-segments  are 
drawn  to  the  endpoints  of  one-  side,  the  sum  of  these  line- 
segments  is  less  than  the  sum  of  the  other  two  sides. 

[348] 

479.  In  the  same  or  in  equal  circles  unequal  chords 
are  unequally  distant  from  the  center,  the  shorter  chord 
lying  at  the  greater  distance;  and  the  converse  of  this 
theorem.     [349] 

480.  If  two  sides  of  one  triangle  are  equal  to  two  sides 
of  another  triangle,  but  the  angle  included  between  the 
two  sides  of  the  first  is  greater  than  the  angle  included 
between  the  corresponding  sides  in  the  second,  then  the 
third  side  in  the  first  is  greater  than  the  third  side  in  the 
second;  and  the  converse  of  this  theorem.     [350] 

481.  In  the  same  or  in  eo  al  circles  the  arcs  sub- 
tended by  unequal  chords  are  unequal  in  the  same  order 
as  the  chords;   and  the  converse  of  this  theorem.     [351] 

Loci.    Concurrent  Lines 

482.  The  locus  of  points  in  a  plane  equidistant  from 
two  given  points  is  the  perpendicular  bisector  of  the  seg- 
ment joining  these  points.     [407] 

483.  The  locus  of  points  in  a  plane  which  are  within 
an  angle  and  equidistant  from  its  sides  is  the  bisector  of 
the  angle.     [408]. 


344  THIRD-YEAR  MATHEMATICS 

484.  The  locus  of  points  in  a  plane  at  a  given  distance 
from  a  given  point  is  the  circle  whose  center  is  the  given 
point  and  whose  radius  is  equal  to  the  given  distance.    [409] 

485.  The  locus  of  points  in  a  plane  at  a  given  distance 
from  a  given  line  consists  of  a  pair  of  lines  parallel  to  the 
given  line  and  the  given  distance  from  it.     [410] 

486.  The  locus  of  points  in  space  equidistant  from  all 
points  on  a  circle  is  the  line  perpendicular  to  the  plane  of 
the  circle  at  the  center.     [411] 

487.  The  locus  of  points  in  space  equidistant  from  two 
given  points  is  the  plane  bisecting  the  segment  joining 
these  points  and  perpendicular  to  it.     [412] 

488.  The  locus  of  points  within  a  diedral  angle  equi- 
distant from  the  faces  is  the  plane  bisecting  the  angle. 

[413] 

Concurrent  Lines 

489.  The  medians  of  a  triangle  are  concurrent.     [417] 

490.  The  perpendicular  bisectors  of  the  sides  of  a 
triangle  are  concurrent  in  a  point  equidistant  from  the 
vertices  of  the  triangle.     [419] 

491.  The  bisectors  of  the  angles  of  a  triangle  are  con- 
current in  a  point  which  is  equidistant  from  the  sides  of 
the  triangle.     [421] 

492.  The  three  altitudes  of  a  triangle  are  concurrent. 

[422] 

Areas 

493.  Parallelograms  having  equal  bases  and  equal 
altitudes  are  equal.     [459] 

494.  The  area  of  a  rectangle  is  equal  to  the  product 
of  the  base  and  the  altitude, 

R  =  b-h.    [481] 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     345 

495.  The  area  of  a  parallelogram  is  equal  to  the 
product  of  the  base  and  the  altitude, 

P  =  b>h.    [482] 

496.  The  area  of  a  triangle  is  equal  to  one-half  the 
product  of  the  base  and  altitude, 

A  =  \b-h.    [465] 

497.  The  area  of  a  triangle  is  equal  to  one-half  the  j 
product  of  two  sides  by  the  sine  of  the  included  angle, 

A  =  \ab  sin  C.     [466] 

498.  The  area  of  a  triangle  is  equal  to  one-half  the 
perimeter  times  the  radius  of  the  inscribed  circle, 

A  =  lp'r=sr.    [467] 

499.  The  area  of  a  triangle  is  equal  to  the  product  of 
the  three  sides  divided  by  four  times  the  radius  of  the 
circumscribed  circle, 

A  =  <£.    [468] 

500.  The  area  of  a  triangle  is  equal  to 


A  =  Vs(s-a)(s-b){s-c).    [469] 

501.  The  area  of  an  equilateral  triangle  is  one-fourth 
the  square  of  a  side  times  the  square  root  of  3, 

4  =  |V3.     [471] 

502.  The  area  of  a  trapezoid  is  equal  to  one-half  the 
product  of  the  altitude  by  the  sum  of  the  bases, 

T  =  \h(bi+b$.     [483] 


346  THIRD-YEAR  MATHEMATICS 

503.  The  area  of  a  regular  inscribed  polygon  is  equal 
to  the  product  of  one-half  the  perimeter  and  the  apothem. 
[484] 

504.  The  area  of  a  regular  circumscribed  polygon  is 
equal  to  the  product  of  one-half  the  perimeter  and  the 
radius.     [485] 


*-/y«fc  R'73fi 


505.  The  area  of  a  circle  is  one-half  the  product  of  the 
length  of  the  circle  and  the  radius,  i.e., 

A  =  \cr.    [489] 

506.  The  area  of  a  circle  is  given  by  the  formula 

A  =  -nr\    [489] 

507.  The  area  of  a  sector  is  given  by  the  formula 

A  =  \a'r.     [490] 

508.  The  area  of  a  segment  of  a  circle  is  given  by  the 
formulas 


H-'-U-t 


or  A  =  l<*'r-\r2  sin  x.     [491] 


Proportionality  of  Areas 

509.  In  a  proportion  the  product  of  the  means  is  equal 
to  the  product  of  the  extremes.     (259) 

510.  The  areas  of  two  rectangles  are  in  the  same  ratio 
as  the  products  of  their  dimensions.     (260) 

511.  Two  rectangles  having  equal  bases  are  in  the 
same  ratio  as  the  altitudes.     (261) 

512.  Two  rectangles  having  equal  altitudes  are  in  the 
same  ratio  as  the  bases.     (262) 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     347 

513.  The  areas  of  parallelograms  are  in  the  same  ratio 

as  the  products  of  the  bases  and  altitudes.     (263) 

i 

514.  The  areas  of  triangles  are  in  the  same  ratio  as 

the  products  of  the  bases  and  altitudes.     (264) 

515.  The  areas  of  parallelograms  having  equal  bases 
are  in  the  same  ratio  as  the  altitudes.     (265) 

516.  The  areas  of  triangles  having  equal  bases  are  in 
the  same  ratio  as  the  altitudes.     (266) 

517.  The  areas  of  two  similar  triangles  are  to  each 
other  as  the  squares  of  any  two  corresponding  sides.     [497] 

518.  The  areas'  of  two  similar  polygons  are  to  each 
other  as  the  squares  of  two  corresponding  sides.     [498] 

Lines  and  Planes  in  Space 

519.  The  following  conditions  determine  the  position 
of  a  plane  in  space : 

1.  A  straight  line  and  a  point  not  in  that  line. 

2.  Three  points  not  in  the  same  straight  line. 

3.  Two  intersecting  straight  lines. 

4.  Two  parallel  straight  lines.     [139] 

520.  If  two  planes  intersect,   the  intersection  is   a 
straight  line.     [143] 

521.  Two  planes  perpendicular  to  the  same  line  are 
parallel.     [178] 

522.  If  two  parallel  planes  are  cut  by  a  third  plane, 
the  intersections  are  parallel.     [179] 

523.  Parallel    line-segments   intercepted    by    parallel 
planes  are  equal.     [180] 


348  THIRD-YEAR  MATHEMATICS 

524.  If  three  or  more  parallel  planes  are  cut  by  two 
transversals,  the  corresponding  segments  of  the  trans- 
versals are  in  proportion.     [181] 

525.  The  projection  upon  a  plane,  of  a  straight  line 
not  perpendicular  to  the  plane,  is  a  straight  line.     [355] 

526.  The  projection  upon  a  plane,  of  a  straight  line 
perpendicular  to  the  plane,  is  a  point.     [356] 

527.  The  acute  angle  formed  by  a  given  line  and  its 
projection  upon  a  plane  is  smaller  than  the  angle  which 
it  makes  with  any  other  line  in  the  plane  passing  through 
the  point  of  intersection  of  the  given  line  and  the  plane. 
[357] 

528.  The  perpendicular  is  the  shortest  distance  from  a 
point  to  a  plane.     [342] 

529.  Oblique  lines  drawn  from  a  point  to  a  plane, 
meeting  the  plane  at  points  equidistant  from  the  foot  of 
the  perpendicular,  are  equal.     [344] 

530.  Oblique  lines  drawn  from  a  point  to  a  plane, 
meeting  the  plane  at  points  unequally  distant  from  the 
foot  of  the  perpendicular,  are  unequal,  the  more  remote 
being  the  greater.     [345] 

531.  Equal  oblique  lines  drawn  from  a  point  to  a  plane 
meet  the  plane  at  points  equidistant  from  the  foot  of  the 
perpendicular.     [346] 

532.  Of  two  unequal  oblique  lines  drawn  from  a  point 
to  a  plane  the  greater  meets  the  plane  at  the  greater 
distance  from  the  foot  of  the  perpendicular.     [347] 

533.  If  a  line  is  perpendicular  to  each  of  two  inter- 
secting lines,  it  is  perpendicular  to  the  plane  determined 
by  these  lines.     [364] 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     349 

534.  All  the  perpendiculars  to  a  given  line  at  a  given 
point  lie  in  a  plane  perpendicular  to  the  given  line  at  the 
point.     [366] 

535.  Only  one  plane  can  be  constructed  perpendicular 
to  a  given  line  at  a  given  point.     [367] 

536.  Only  one  plane  can  be  constructed  perpendicular 
to  a  given  line  from  a  point  outside  of  the  line.     [368] 

537.  Only  one  line  can  be  constructed  perpendicular 
to  a  given  plane  at  a  given  point.     [370] 

538.  From  a  point  outside  of  a  given  plane  only  one 
line  can  be  constructed  perpendicular  to  the  plane.     [372] 

539.  Lines  perpendicular  to  a  plane  are  parallel.     [373] 

540.  If  one  of  two  parallel  lines  is  perpendicular  to  a 
plane,  the  other  is  perpendicular  to  the  same  plane.     [374] 

541.  Two  lines  parallel  to  the  same  line  are  parallel  to 
each  other.     [375] 

542.  If  two  lines  are  parallel,  a  plane  containing  one 
of  them  and  not  the  other  is  parallel  to  the  other.     [376] 

543.  If  one  of  two  parallel  planes  is  perpendicular  to  a 
line,  the  other  is  also.     [377] 

544.  If  two  intersecting  lines  are  parallel  to  a  given 
plane,  their  plane  is  parallel  to  the  given  plane.     [378] 

545.  If  two  angles  not  in  the  same  plane  have  their 
sides  parallel  and  running  in  the  same  direction,  the  angles 
are  equal  and  their  planes  are  parallel.     [379] 

546.  All  plane  angles  of  a  diedral  angle  are  equal. 
[380] 

547.  If  two  diedral  angles  are  equal,  their  plane  angles 
are  equal.     [381] 


350  THIRD-YEAR  MATHEMATICS 

548.  Two  diedral  angles  are  equal  if  the  plane  angles 
are  equal.     [381] 

549.  If  a  line  is  perpendicular  to  a  plane,  every  plane 
through  this  line  is  perpendicular  to  the  plane.     [382] 

550.  If  two  planes  are  perpendicular  to  each  other,  a 
line  drawn  in  one  of  them  perpendicular  to  the  intersection 
is  perpendicular  to  the  other.     [383] 

551.  If  two  planes  are  perpendicular  to  each  other,  a 
line  perpendicular  to  one  of  them  at  a  point  of  the  inter- 
section must  lie  in  the  other.     [383] 

552.  If  from  a  point  in  one  of  two  perpendicular 
planes  a  line  is  drawn  perpendicular  to  the  other,  it  must 
lie  in  the  first  plane.     [383] 

553.  If  a  plane  is  perpendicular  to  each  of  two  planes, 
it  is  perpendicular  to  their  intersection.     [384] 

554.  Through  a  line  not  perpendicular  to  a  given  plane 
one  plane  and  only  one  may  be  passed  perpendicular  to  the 
given  plane.     [385] 

555.  The  section  of  a  sphere  made  by  a  plane  is  a  circle. 
[389] 

556.  The  axis  of  a  circle  passes  through  the  center. 
[390] 

557.  The  diameter  of  a  sphere  passing  through  the 
center  of  a  circle  is  perpendicular  to  the  plane  of  the  circle. 
[390] 

558.  All  great  circles  of  a  sphere  are  equal.     [390] 

559.  Every  great  circle  bisects  the  surface  of  the 
sphere.     [390] 

560.  Through  two  points  on  the  surface  of  a  sphere, 
not  the  endpoints  of  a  diameter,  only  one  great  circle  can 
be  drawn.     [390] 


ASSUMPTIONS  AND  THEOREMS  OF  GEOMETRY     351 

561.  All  points  on  a  circle  of  a  sphere  are  equidistant 
from  its  poles.     [392] 

562.  The  polar  distance  of  a  great  circle  is  a  quadrant. 
[395] 

563.  If  a  point  on  the  surface  of  a  sphere  is  at  the 
distance  of  a  quadrant  from  each  of  two  given  points  on  the 
surface,  it  is  a  pole  of  the  great  circle  passing  through  the 
given  points.     [396] 

564.  The  intersection  of  the  surfaces  of  two  spheres  is 
a  circle  whose  plane  is  perpendicular  to  the  line  of  centers 
of  the  spheres,  and  whose  center  is  in  that  line.     [397] 

565.  A  plane  tangent  to  a  sphere  is  perpendicular  to 
the  radius  at  the  point  of  contact.     [399] 

566.  A  plane  perpendicular  to  a  radius  of  a  sphere  at 
the  outer  extremity  is  tangent  to  the  sphere.     [400] 

Constructions 

567.  Through  a  given  point  in  a  given  line  pass  a  plane 
perpendicular  to  the  given  line.     [365] 

568.  From  a  given  point  outside  of  a  given  line  con- 
struct a  plane  perpendicular  to  the  given  line.     [368] 

569.  At  a  given  point  in  a  given  plane  construct  a  per- 
pendicular to  the  plane.     [369] 

570.  From  a  point  outside  of  a  plane  construct  a  line 
perpendicular  to  the  plane.     [371] 

571.  To  pass  a  plane  perpendicular  to  a  given  plane, 
that  shall  contain  a  line  not  perpendicular  to  the  given 
plane.     [385] 


LOGARITHMIC  AND  TRIGONOMETRIC 

TABLES  AND  MATHEMATICAL 

FORMULAS 


# 


CONTENTS 


TABLE  PAGE 

I.  Common  Logakithms  of  Numbers      ...  1 

II.  Common  Logarithms  of  the  Trigonometric 

Functions 21 

III.  Values  of  the  Natural  Trigonometric 

Functions 73 

IV.  Tables  of  Powers  and  Roots       ....  99 

V.  Formulas 108 

VI.  Equivalents  and  Logarithms  of  Important 

Constants '  .  Ill 

VII.  Reductions 115 


TABLE  I 

Common  Logarithms  of  Numbers  from  1  to  10000 
to  Five  Decimal  Places 


I] 

1000 — Common  Logarithms  of  Numbers 

1500    3 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

.PP 

100 

00 

000 

043 

087 
»«18 

*30 

;  17-& 

217 

260 

303 

346 

389 

101 

432 

475* 

604 

647 

689 

732 

775 

817 

102 

860 

903 

945 

988 

*030 

*072 

*115 

*157 

*199 

*242 

44  43  4*2 

103 

01 

284 

326 

368 

4m 

828 

*453« 

494 

536 

578 

620 

662 

104" 

703 

745 

787 

870 

912 

953 

995 

*036 

*078 

1 

a 

3 

4.4  4.3  4X 
8.8  8.6  8.4 
13.2  12.9  12.6 

105 

02 

119 

160 

202 

243 

284 

325 

366 

407 

449 

490 

10Q 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

4 

17.6  17.2  16.8 

107 

938 

979 

*019 

*060 

*100 

*141 

*181 

*222 

*262 

*302 

5 

22.0  21.5  21.0 

108 

03 

342 

383 

423 

463 

503 

543 

583 

623 

663 

703 

6 

26.4  25.8  25.2 

109 

743 

782 

822 

862 

902 

941 

981 

*021 

*060 

*100 

7 

8 

30.8  30.1  29.4 
35.2  34,4  33.6 
39.6  3S;7  3%.*8 

110 

04 

139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

© 

111 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

112 

922 

961 

999 

*038 

*077 

*115 

*154 

*192 

*231 

*269 

41  40  39 

•113 

05 

308 

346 

385i  423 

461 

500 

538 

576 

614 

652 

1 

4  1  4  0  3  9 

114 

690 

729 

767 

805 

843 

881 

918 

956 

994 

*032 

2 

8.2  8.0  7.8 

115 

06  070 

108 

145 

183 

221 

258 

296 

333 

371 

408 

3 

4 

12.3  12.0  11.7 

16.4  16.0  15.6 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

5 

■20.5  20.0  19.5 

117 

819 

856 

893 

930 

967 

*004 

*041 

*078 

*115 

*151 

6 

24.6  24.0  23.4 

118 

07 

188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

7 

28  7  28  0  27  3 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

8 

32.8  32.0  31.2 

120 

918 

954 

990 

*027 

*063 

*099 

*135 

*171 

*207 

*243 

Q 

36.9  36.0  35.1 

121 

OS 

279 

314 

350 

386 

422 

458 

m 

529 

565 

600 

38  37  36 

122 

636 

^672 

707 

743 

778 

814 

884 

920 

955 

123 

991 

*026 

*061 

*096 

*132 

*167 

*202 

*237 

*272 

*307 

1 

6. 8  3.7  3.6 

124 

09 

342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

2 
3 

7.6  7.4  7.2 
11  4  11  1  10  8 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968 

*003 

4 

15.2  14.8  14.4 

126 

10 

037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

5 

19.0  18.5  18.0 

127 

380 

415 

449 

483 

517 

551 

585 

619 
958 

653 

687 

6 

22.8  22.2  21.6 

128 

721 

755 

789 

823 

857 

890 

924 

992 

*025 

7 

26.6  25.9  25.2 

129 

11 

059 

093 

126 

160 

193 

227 

261 

294 

.327 

361 

8 
9 

30.4  29.6  28.8 
34.2  33.3  32.4 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 

131 

727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 

35  34  33 

132 

12 

057 

090 

123 

156 

189 

222 

254 

287 

320 

352 

1 

3.5  3.4  3.3 

133 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

2 

7.0  6.8  6.6 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

*001 

3 
4 

10.5  10.2  9.9 
14.0  13.6  13.2 

135 

13 

033 

066 

098 

130 

162 

194 

226 

258 

290 
609 

322 

5 

17.5  17.0  16.5 

136 

354 

386 

418 

450 

481 

513 

545 

577 

640 

6 

21.0  20.4  19.8 

137 

672 

1  704 

735 

767 

799 

830 

862 

893 

925 

956 

7 

24.5  23.8  23.1 

138 

988 

*019 

*051 

*082 

*114 

*145 

*176 

*208 

*239 

*270 

8 

28.0  27.2  26.4 

139 

14 

301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

9 

31.5  30.6  29.7 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

32  31  30 

141 

922 

953 

983 

*014 

*045 

*076 

*106 

*137 

*168 

*198 

1 

3.2  3.1  3.0 
6.4  6.2  6.0 
9.6  9.3  9.0 
12.8  12.4  12.0 

142 

15 

229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

2 
3 

4 

143 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

144 

836 

866 

897 

927 

957 

987 

*017 

*047 

*077 

*107 

145 

16 

137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

5 
6 

7 
8 
9 

16.0  15.5  15.0 
19.2  18.6  18.0 
22.4  21.7  21.0 
25.6  24.8  24.0 
28.8  27.9  27.0 

146 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

148 

17 

026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

49 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

150 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

4    1500— 

Common  Logarithms  of  Numbers — 

2000   [I 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

150 

17  609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

15V 

898 

926 

955 

984 

*013 

*041 

*070*099 

*127 

*156 

29   28 

152 

18  184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

153 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

1 

2.9  2.8 

154 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*005 

2 
3 

5.8  5.6 

8.7  8.4 

155 

19  033 

061 

089 

117 

145 

173 

201 

229 

n257 

285 

4 

11.6  11.2 

156 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

5 

14.5  14.0 

157 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

6 

17.4  16.8 

158 

866 

893 

921 

948 

976 

*003 

*030  *058 

*085 

*112 

7 

20.3  19.6 

159 

20  140 

167 

194 

222 

249 

276 

303 

330 

358 

385 

8 
9 

23.2  22.4 
26.1  25.2 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

161 

683 

710 

737 

763 

790 

817 

844 

871 

898 

925 

27   26 

162 

952 

978 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

163 

21  219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

1 

2.7  2.6 

164 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

2 

a 

5.4  5.2 

8.1  7.8 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

4 

10.8  10.4 

166 

22  Oil 

037 

063 

089 

115 

141 

167 

194 

220 

246 

5 

13.5  13.0 

167 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

6 

16.2  15.6 

168 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

7 

18.9  18.2 

169 

789 

814 

840 

866 

891 

917 

943 

968 

994 

*019 

8 
9 

21.6  20.8 
24.3  23.4 

170 

23  045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

171 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

25 

172 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 

173 

805 

830 

855 

880 

905 

930 

955 

980 

*005 

*030 

1 

2.5 

174 

24  055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

2 
3 
4 

5.0 
7.5 

10.0 

175 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

176 

551 

576 

601 

625 

650 ' 

674 

699 

724 

748 

773 

5 

12.5 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993 

*018 

6 

15.0 

178 

25  042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

7 

17.5 

179 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 

8 
9 

20.0 
22.5 

ISO 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

181 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24   23 

182 

26  007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

183 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

1 

2.4  2.3 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

2 
3 

4.8  4.6 
7.2  6.9 

185 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

4 

9.6  9.2 

185 

951 

975 

998 

*021 

*045 

*068 

*091 

*114 

*138 

*161 

5 

12.0  11.5 

187 

27  184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

G 

14.4  13.8 

188 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

7 

16.8  16.1 

189 

646 

669 

692 

715 

738 

761 

784 

807 

830 

852 

8 

9 

19.2  18.4 
21.6  20.7 

190 

875 

898 

921 

944 

967 

989 

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191 

28  103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

22   21 

192 

330 

353 

375 

398 

421 

443  466 

488 

511 

533 

193 

556 

578 

601 

623 

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713 

735 

758 

1 

2.2  2.1 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

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2 

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4.4  4.2 
6.6  6.3 

195 

29  003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

4 

8.8  8.4 

196 

226 

248 

270 

292 

314 

336 

358 

380 

403 

425 

6 

11.0  10.5 

197 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

6 

13.2  12.6 

198 

667 

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710 

732 

754 

776 

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820 

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7 

15.4  14.7 

199 

885 

907 

929 

951 

973 

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8 
9 

17.6  <3.b 
19.8  18.9 

200 

30  103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

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; 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

200 

30  103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

201 

320 

341 

363 

384 

406 

428 

449 

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492 

514 

22  21 

202 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

203 

750 

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792 

814 

835 

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878 

899 

920 

942 

1 

2.2  2.1 

204 

963 

984 

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2 
3 

4.4  4.2 
6.6  6.3 

205 

31  175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

4 

8.8  8.4 

206 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

5 

11.0  10.5 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

6 

13.2  12.6 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

7 

15.4  14.7 

209 

32  015 

035 

056 

077 

098 

118 

139 

160 

181 

201 

8 
9 

17.6  16.8 
19.8  18.9 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

211 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 

20 

212 

634 

654 

675 

695 

715 

736 

756 

777 

797|  818 

213 

838 

858 

879 

899 

919 

940 

960 

980 

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1 

2.0 

214 

33  041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

2 
3 

4- 

4.0 
6.0 
8.0 

215 

244 

264 

284 

304 

325 

345 

365 

385 

405 

425 

216 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

5 

10.0 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

6 

12.0 

218 

846 

866 

885 

905 

925 

945 

965 

985 

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7 

14.0 

219 

34  044 

064 

084 

104 

124 

143 

163 

183 

203 

223 

8 
9 

16.0 
18.0 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

19 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

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1 

1.9 

224 

35  025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

2 
3 
4 

3.8 
5.7 
7.6 

225 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

226 

411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

5 

9.5 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

6 

11.4 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946 

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7 

13.3 

229 

984 

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8 
9 

15.2 
17.1 

230 

36  173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

231 

361 

380 

399 

418 

436 

455 

■  474 

493 

511 

530 

18 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

233 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

1 

1.8 

234 

922 

940 

959 

977 

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2 
3 
4 

3.6 
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235 

37  107 

125 

144 

162 

181 

199 

218 

236 

254 

273 

236 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

5 

9.0 

237 

475 

493 

511 

530 

548 

566 

585 

603 

621 

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6 

10.8 

238 

658 

676 

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712 

731 

749 

767 

785 

803 

822 

7 

12.6 

239 

840 

858 

876 

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912 

931 

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967 

985 

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8 
9 

14.4 
16.2 

240 

38  021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

241 

202 

220 

238 

256 

274 

292 

31Q 

328 

346 

364 

17 

242 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

243 

56, 1 

578 

596 

614 

632 

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668 

686 

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1 

1.7 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

2 
3 
4 

3.4 
5.1 
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245 

917 

934 

952 

970 

987 

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246 

39  094 

111 

129 

146 

164 

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217 

235 

252 

5 

8.5 

247 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

6 

10.2 

248 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

7 

11.9 

249 

620 

637 

655 

672 

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707 

724 

742 

759 

777 

8 
9 

13.6 
15.3 

250 

794 

811 

829 

846 

863 

881 

898 

915 

933 

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N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

6    2500 — Common  Logarithms  of 

Numbers — 

3000    [I 

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0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

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250 

39  794 

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881 

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915 

933 

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251 

967 

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18 

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40  140 

157 

175 

192 

209 

226 

243 

261 

278 

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253 

312 

329 

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364 

381 

398 

415 

432 

449 

466 

1 

1.8 

254 

483 

500 

518 

535 

552 

569 

586 

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620 

637 

2 
3 
4 

3.6 
5.4 

7.2 

255 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

256 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

5 

9.0 

257 

993 

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6 

10.8 

258 

41  162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

7 

12.6 

259 

330 

347 

363 

380 

397 

414 

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514 

531 

547 

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614 

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261 

664 

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814 

17 

262 

830 

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880 

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913 

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963 

979 

263 

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1 

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264 

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177 

193 

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226 

243 

259 

275 

292 

308 

2 
3 
4 

3.4 
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265 

325 

341 

357 

374 

390 

406 

423 

439 

455 

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266 

488 

504 

521 

537 

553 

570 

586 

602 

619 

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5 

8.5 

267 

651 

667 

684 

700 

716 

732 

749 

765 

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6 

10.2 

268 

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830 

846 

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878 

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911 

927 

943 

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7 

11.9 

269 

975 

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8 
9 

13.6 
15.3 

270 

43  136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

271 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

16 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

273 

616 

632 

648 

664 

680 

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712 

727 

743 

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1 

1.6 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

2 
3 
4 

3.2 
4.8 
6.4 

275 

933 

949 

965 

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276 

44  091 

107 

122 

138 

154 

170 

185 

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217 

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5 

8.0 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

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6 

9.6 

278 

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420 

436 

451 

467 

483 

498 

514 

529 

545 

7 

11.2 

279 

560 

576 

592 

607 

623 

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669 

685 

700 

8 
9 

12.8 
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731 

747 

762 

778 

793 

809 

824 

840 

855 

281 

871 

886 

902 

917 

932 

948 

963 

979 

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15 

282 

45  025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

1 

1.5 

284 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

2 
3 

4 

3.0 
4.5 
6.0 

285 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

288 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

5 

7.5 

287 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

6 

9.0 

288 

939 

954 

969 

984 

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7 

10.5 

289 

46  090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

8 
9 

12.0 
13.5 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

14 

292 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

293 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

1 

1.4 

294 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

2 
3 
4 

2.8 
4.2 
5.6 

295 

982 

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296 

47  129 

144 

159 

173 

188 

202  217 

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5 

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297 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

6 

8.4 

298 

422 

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451 

465 

480 

494 

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524 

538 

553 

7 

9.8 

299 

567 

582 

596 

611 

625 

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9 

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300 

712 

727 

741 

756 

770 

784 

799 

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828 

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0 

1 

2 

3 

4 

5 

6 

7 

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9 

PP 

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3000 — Common  Logarithm 

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0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

300 

47  712 

727 

741 

756 

770 

784 

799 

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828 

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301 

857 

871 

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900 

914 

929 

943 

958 

972 

986 

302 

48  001 

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029 

044 

058 

073 

087 

101 

116 

130 

303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

15 

304 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 

1 

2 

1.5 

3.0 

305 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

306 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

3 

4.5 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

4 

6.0 

308 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

5 

7.5 

309 

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7 
8 

9.0 
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310 

49  136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

311 

276 

290 

304 

318 

332 

346 

360 

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9 

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312 

415 

429 

443 

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471 

485 

499 

513 

527 

541 

313 

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568 

582 

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610 

624 

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651 

665 

679 

314 

693 

707 

721 

734 

748 

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790 

803 

817 

315 

831 

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872 

886 

900 

914 

927 

941 

955 

14 

316 

969 

982 

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1 

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317 

50  106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

2 

2.8 

318 

243 

256 

270 

284 

297 

311 

325 

338 

352 

365 

3 

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319 

379 

393 

406 

420 

433 

447 

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4 

5!6 

320 

515 

529 

542 

556 

569 

583 

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610 

623 

637 

5 
6 

7.0 

8.4 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

7 

9^8 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

8 

11.2 

323 

920 

934 

947 

961 

974 

987 

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9 

12^6 

324 

51  055 

068 

081 

095 

108 

121 

135 

148 

162 

175 

325 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 

326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 

327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

13 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

2 

1.3 
2.6 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

3 

3.9 

331 

983 

996 

*009 

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4 

5.2 

332 

52  114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

5 

6.5 

333 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

6 

7.8 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

7 
8 

9.1 

10.4 

335 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 

9 

11.7 

336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 

337 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 

338 

892 

905 

917 

930 

943 

956 

969 

982 

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339 

53  020 

033 

046 

058 

071 

084 

097 

110 

122 

135 

12 

340 

148 

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173 

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212 

224 

237 

250 

263 

341 

275 

288 

301 

314 

326 

339 

352 

364 

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1 

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342 

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2 

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343 

529 

542 

555 

567 

580 

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344 

656 

668 

681 

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719 

732 

744 

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769 

4 
5 
6 

4.8 
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345 

782 

794 

807 

820 

832 

845 

857 

870 

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895 

346 

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920 

933 

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970 

983 

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7 

8.4 

347 

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070 

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108 

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145 

8 

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348 

158 

170 

183 

195 

208 

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258 

270 

9 

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283 

293 

307 

320 

332 

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357 

370 

382 

394 

350 

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419 

432 

444 

456 

469 

481 

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0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

8    3500 — Common  Logarithms  of  Numbers — 4000 

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0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

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350 

54  407 

419 

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351 

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543 

555 

568 

580 

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617 

630 

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352 

654 

667 

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691 

704 

716 

728 

741 

753 

765 

353 

777 

790 

802 

814 

827 

839 

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864 

876 

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13 

354 

900 

913 

925 

937 

949 

962 

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1.3 

355, 

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108 

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2 

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145 

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255 

3 

3.9 

357 

267 

279 

291 

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315. 

328 

340 

352 

364 

376 

4 

5.2 

358 

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400 

413 

425 

437 

449 

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5 

6.5 

359 

509 

522 

534 

546 

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570 

582 

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6 

7 

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727 

739 

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361 

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763 

775 

787 

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9 

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362 

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56  110 

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253 

265 

277 

289 

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324 

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419 

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367 

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514 

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549 

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368 

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667 

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2 

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3 
4 

3.6 

4.8 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

5 

6.0 

371 

937 

949 

961 

972 

984 

996 

*008 

*019 

*031 

*043 

6 

7.2 

372 

57  054 

066 

078 

089 

101 

113 

124 

136 

148 

159 

7 

8.4 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

8 

9.6 

374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 

9 

10.8 

375 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 

376 

519 

530 

542 

553 

565 

576 

588 

600 

611 

623 

377 

634 

646 

657 

669 

680 

692 

703 

715 

726 

738 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

11 

379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

1 

1.1 

880 

978 

990 

*001 

*013 

*024 

*035 

*047 

*058 

*070 

*081 

2 

2.2 

381 

58  092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

3 

3.3 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

4 

4.4 

383 

320 

331 

343 

354 

365 

377 

388 

399 

410 

422 

5 

5.5 

384 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

6 

7 

6.6 

7.7 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

8 

8.8 

386 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 

9 

9.9 

387 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 

388 

883 

894 

906 

917 

928 

939 

950 

961 

973 

984 

389 

995 

*006 

*017 

*028 

*040 

*051 

*062 

*073 

♦084 

♦095 

390 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

10 

391 

218 

229 

240 

251 

262 

273 

284 

295 

306 

318. 

392 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 

1 

l.U 

393 

439 

450 

461 

472 

483 

494 

506 

517 

528 

539 

2 

2.0 

394 

550 

561 

572 

583 

594 

605 

616 

627 

638 

649 

3 
4 

3.0 
4.0 

395 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

5 

5.0 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

6 

6.0 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

7 

7.0 

398 

988 

999 

♦010 

♦021 

♦032 

♦043 

*054 

*065 

*076 

*086 

8 

8.0 

399 

60  097 

108 

119 

130 

141 

152 

163 

173 

184 

195 

9 

9.0 

400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

I] 

4000 — Common  Logarithm 

s  of  Numbers— 4500    9 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

401 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 

405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

n 

407 

959 

970 

981 

991 

*002 

*013 

*023 

*034 

*045 

*055 

408 

61  066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

i 

1.1  • 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

2 
3 
4 

2.2 
3.3 
4.4 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

5 

5.5 

412 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

6 

6.6 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

7 

7.7 

414 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

8 
9 

8.8 
9.9 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

416 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*003 

417 

62  014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

418 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 

419 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

421 

428 

439 

449 

459 

469 

480 

490 

500 

511  521 

10 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613  624 

423 

634 

644 

655 

665 

675 

685 

696 

706 

716  726 

1 

1.0 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

2 
3 
4 

2.0 
3.0 
4.0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

426 

941 

951 

961 

972 

982 

992 

*002 

*012 

*022 

*033 

5 

5.0 

427 

63  043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

6 

6.0 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

7 

7.0 

429 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

8 
9 

8.0 
9.0 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

431 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 

432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

436 

949 

959 

969 

979 

988 

998 

*008 

*018 

*028  *038 

9 

437 

64  048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

1 

0.9 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

2 
3 
4 

1.8 
2.7 
3.6 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

441 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

5 

4.5 

442 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

6 

5.4 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

7 

6.3 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

8 
9 

7.2 

8.1 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 

446 

933 

943 

953 

963 

972 

982 

992 

♦002 

*011 

*021 

447 

65  031 

040 

050 

060 

070 

079 

089 

099 

108 

118 

448 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 

449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

10 

4500 — Common  Logarithm 

s  of  Numbers — 

5000   [I 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

450 

65  321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 

452 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 

453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

454 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 

455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

10 

457 

992 

*001 

*011  *020  *030 

*039 

*049 

*058 

*068 

*077 

458 

66  087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

i 

1.0 

459 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

2 
3 
4 

2.0 
3.0 
4.0 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

461 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

5 

5.0 

462 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

6 

6.0 

463 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

7 

7.0 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

8 
9 

8.0 
9.0 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

467 

932 

941 

950 

960 

969 

978 

987 

997 

*006 

*015 

468 

67  025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

471 

302 

311 

321 

330 

339 

348 

357 

367 

376 

385 

i     9 

472 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

473 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

1 

0.9 

474 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

2 
3 
4 

1.8 
2.7 
3.6 

475 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

5 

4.5 

477 

852 

861 

870 

879  888 

897 

906 

916 

925 

934 

6 

5.4 

478 

943 

952 

961 

970 

979 

988 

997 

*006 

*015 

*024 

7 

6.3 

479 

68  034 

043 

052 

061 

070 

079 

088 

097 

106 

115 

8 
9 

7.2 
8.1 

ISO 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 

483 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

8 

487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

488 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

1 

0.8 

489 

931 

940 

949 

958 

966 

975 

984 

993 

*002 

*01.1 

2 
3 
4 

1.6 

2.4 
3.2 

490 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

5 

4.0 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

6 

4.8 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

7 

5.6 

494 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

8 
9 

6.4 

7.2 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

497 

636 

644 

653 

662 

671 

679 

688 

697 

705 

714 

498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 

500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

I] 

5000 — Common  Logarithms  of  Numbers — 

550C 

1    11 

N 

0 

1 

2 

3 

4 

5 

6  7 

8 

9 

PP 

500 

69  897 

906|  914 

923 

932 

940 

949  958 

966 

975 

501 

984 

992  *001 

*010  *018 

*027 

*036  *044 

*053 

*062 

502 

70  070 

0791  088 

096 

105 

114 

122 

131 

140 

148 

503 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 

504 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 

505 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

9 

507 

501 

500 

518 

526 

535 

544 

552 

561 

569 

578 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

1 

0.9 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

2 
3 

i  4 

1.8 
2.7 
3.6 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

511 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

5 

4.5 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995  *003 

6 

5.4 

513 

71  012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

7 

6.3 

514 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 

8 
9 

7.2 
8.1 

515 

181 

189 

198 

206 

214 

223 

231 

24l0 

248 

257 

516 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 

517 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 

518 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 

519 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 

520 

600 

609 

617 

625 

634 

642 

550 

659 

667 

675 

521 

684 

692 

700 

709 

717 

725 

734 

742 

750  759 

8 

522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

1 

0.8 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999 

*008 

2 
3 
4 

1.6 
2.4 
3.2 

525 

72  016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

5 

4.0 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

6 

4.8 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

7 

5.6 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

8 
9 

6.4 
7.2 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

531 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 

532 

591 

599 

607 

616 

624 

632 

640  648 

656 

665 

533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 

534 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 

535 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 

536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 

7 

537 

997 

*006 

*014 

*022 

*030 

*038 

*046 

*054 

*062 

*070 

538 

73  078 

086 

094 

102 

111 

119 

127i  135 

143 

151 

1 

0.7 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

2 
3 
4 

1.4 
2.1 
2.8 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

541 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

5 

3.5 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

6 

4.2 

543 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

7 

4.9 

544 

560 

•568 

576 

584 

592 

600 

608 

616 

624 

632 

8 
9 

5.6 
6.3 

545 

640 

648 

656 

664 

672 

679 

687 

695 

703 

711 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 

547 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 

548 

878 

886 

894 

902 

910 

918 

926 

933 

941 

949 

549 

957 

965 

973 

981 

989 

997 

*005 

*013 

*020 

*028 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

12 

5500 — Common  Logarithm 

s  of  Numbers — 

6000    [I 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

551 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 

552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 

553 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 

554 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 

555 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 

556 

507 

515 

523 

531 

539 

547 

554 

562 

570 

578 

557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 

558 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 

559 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

8 

561 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 

562 

974 

981 

989 

997 
074 

*005 

*012 

*020 

*028 

*035 

*043 

1 

0.8 

563 

75  051 

059 

066 

082 

089 

097 

105 

113 

120 

2 

1.6 

564 

128 

136 

143 

151 

159 

166 

174 

182 

189 

197 

3 
4 
5 

2.4 
3.2 
4.0 

565 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

6 

4.8 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

7 

5.8 

568 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

8 

6.4 

569 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 

9 

7.2 

570 

587 

595 

6Q3 

610 

618 

626 

633 

641 

648 

656 

571 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 

572 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 

573 

815 

823 

831 

838 

846 

853 

.861 

868 

876 

884 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

575 

967 

974 

982 

989 

997 

*005 

*012 

*020 

*027 

*035 

576 

76  042 

050 

057 

065 

072 

080 

087 

095 

103  110 

577 

118 

125 

133 

140 

148 

155 

163 

170 

1781  185 

578 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 

579 

26j8 

275 

283 

290 

298 

305 

313 

320 

328 

335 

58© 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

7 

581 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 

582 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

1 

0.7 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

2 

1.4 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

3 
4 
5 

2.1 
2.8 
3.5 

585 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

6 

4.2 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923  930 

7 

4.9 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997  *004 

8 

5.6 

589 

7J_012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

9 

6.3 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

591 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 

592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 

593 

305 

313 

320 

327 

335 

342 

349 

357 

364 

371 

594 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 

595 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 

596 

525 

532 

539 

546 

554 

561 

568 

576 

583 

590 

597 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 

598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

599 

743 

750 

757 

764 

772 

779 

786 

793 

801 

808 

600 

815 

^822 

830 

837 

844 

851 

859 

866 

873 

880 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

I]    6000 — Common  Logarithms  of  Numbers — 6500   13 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8  |  9 

PP 

600 

77  815 

822 

830 

837 

844 

851 

859 

866 

873!  880 

601 

887 

895 

902 

909 

916 

924 

931 

938 

945  952 

602 

960 

967 

974 

981 

988 

996 

*003 

*O10 

*017  *025 

603 

78  032 

039 

046 

053 

061 

068 

075 

082 

089  097 

604 

104 

111 

118 

125 

132 

140 

147 

154 

161  168 

605 

176 

183 

190 

197 

204 

211 

219 

226 

233  240 

606 

247 

254 

262 

269 

276 

283 

290 

297 

305  312 

8 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

1 

0.8 

609 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

2 
3 
4 

1.6 
2.4 
3.2 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

5 

4.0 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

6 

4.8 

613 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

7 

5.6 

614 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 

8 
9 

6.4 
7.2 

615 

888 

895 

902 

909 

916 

923 

930 

937 

944  951 

616 

958 

965 

972 

979 

986 

993 

*000 

*007 

*014  *021 

617 

79  029 

036 

043 

050 

057 

064 

071 

078 

O85  092 

618 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 

619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

621 

309 

316 

323 

330 

337 

344 

351 

358 

365  372 

7 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435  442 

623 

449 

456 

463 

470 

477 

484 

491 

498 

505!  511 

1 

0.7 

624 

518 

525 

532 

539 

546 

553 

560 

567 

574  581 

2 
3 
4 

1.4 
2.1 

2.8 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644  650 

626 

657 

664 

671 

678 

685 

692 

699 

706 

713  720 

5 

3.5 

627 

727 

734 

741 

748 

754 

761 

768 

775 

782  789 

6 

4.2 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851  858 

7 

4.9 

629 

865 

872 

879 

886 

893 

900 

906 

913 

920,  927 

8 
9 

5.6 
6.3 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989!  996 

631 

80  003 

010 

017 

024 

030 

037 

044 

051 

058  065 

632 

072 

079 

085 

092 

099 

106 

113 

120 

127  134 

633 

140 

147 

154 

161 

168 

175 

182 

188 

195  202 

634 

209 

216 

223 

229 

236 

243 

250 

257 

264'  271 

635 

277 

284 

291 

298 

305 

312 

318 

325 

332  339 

635 

346 

353 

359 

366 

373 

380 

387 

393 

400  407 

6 

637 

414 

421 

428 

434 

441 

448 

455 

462 

468  475 

638 

482 

489 

496 

502 

509 

516 

523 

530 

536  543 

1 

0.6 

639 

/  550 

557 

564 

570 

577 

584 

591 

598 

604,  611 

2 
3 
4 

1.2 
1.8 
2.4 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672!  679 

641 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

5 

3.0 

642 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

6 

3.6 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

7 

4.2 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

8 
9 

4.8 
5.4 

645 

956 

963 

969 

976 

983 

990 

996 

*003 

*010*017 

646 

81  023 

030 

037 

043 

050 

057 

064 

070 

077 

084 

647 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 

648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 

650 

291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

14 

6500 — Common  Logari 

thm 

s  of  Numbers — 

7000    [I 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

650 

81  291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

651 

358 

365 

371 

378 

385 

391 

398 

405 

411 

418 

652 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 

653 

491 

498 

505 

511 

518 

525 

531 

538 

544 

551 

654 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 

655 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 

656 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

657 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 

658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

659 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

660 

954 

961 

968 

974 

981 

987 

994 

*000 

*007 

*014 

7 

661 

82  020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

662 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

1 

0.7 

663 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

2 

1.4 

664 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

3 

2.1 

665 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

4 
5 

2.8 
3.5 

666 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

6 

4.2 

667 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

7 

4.9 

668 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

8 

5.6 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

9 

6.3 

67© 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

671 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 

672 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

673 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 

674 

866 

872 

879 

885 

892 

898 

905 

911 

918 

824 

675 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 

676 

995 

*001 

*008 

*014 

*020 

*027 

*033 

*040 

*046 

*052 

677 

83  059 

065 

072 

078 

O85 

091 

097 

104 

110 

117 

678 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

679 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

681 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

6 

682 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

683 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

1 

0.6 

684 

506 

512 

518 

525 

531 

537 

544 

550 

556 

563 

2 
3 
4 

1.2 
1.8 
2.4 

685 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

686 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

5 

3.0 

687 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

6 

3.6 

688 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

7 

4.2 

689 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

8 
9 

4.8 
5.4 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

691 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*004 

692 

84  Oil 

017 

023 

029 

036 

042 

048 

055 

061 

067 

693 

073 

080 

086 

092 

098 

105 

111 

117 

123 

130 

694 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

695 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

696 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

i 

697 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

698 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

699 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

I] 

7000 — Common  Logarithm 

s  of 

Numbers — 

7500   15 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

700 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

701 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 

702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 

703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 

704 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 

7©5 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

706 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 

7 

707 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 

708 

85  003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

1 

0.7 

709 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

2 
3 
4 

1.4 
2.1 
2.8 

71© 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

711 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

5 

3.5 

712 

248 

254 

260 

266 

2?2 

278 

285 

291 

297 

303 

6 

4.2 

713 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

7 

4.9 

714 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

8 
9 

5.6 
6.3 

715 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 

716 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 

717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

718 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

719 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 

720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

6 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

723 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

1 

0.6 

724 

974 

980 

986 

992 

998 

*004 

*010 

*016 

*022 

*028 

2 
3 
4 

1.2 
1.8 
2.4 

725 

86  034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

5 

3.0 

727 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

6 

3.6 

728 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

7 

4.2 

729 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

8 
9 

4.8 
5.4 

730 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

731 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 

732 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 

733 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 

734 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 

735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

736 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

5 

737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

1 

0.5 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

2 
3 
4 

1.0 
1.5 

2.0 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

741 

982 

988 

994 

999 

*005 

*011 

*017 

*023 

*029 

*035 

5 

2.5 

742 

87  040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

6 

3.0 

743 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

7 

3.5 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

8 
9 

4.0 

4.5 

745 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 

746 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 

747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

748 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 

749 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 

750 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

16 

7500 — Common  Logarithms  of  Numbers — 

8000    [I 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

750 

87  506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

751 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 

752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 

753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 

754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 

756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 

757 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 

758 

967 

973 

978 

984 

990 

996 

*001 

*007 

*013 

*018 

759 

88  024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 

761 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 

6 

762 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

1 

0.6 

764 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

2 

1.2 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

3 
4 

1.8 
2.4 

766 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

5 

3.0 

767 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

6 

3.6 

768 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

7 

4.2 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

8 

4.8 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

9 

5.4 

771 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 

772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 

773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

775 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 

776 

986 

992 

997 

*003 

*009 

*014  *020 

*025 

*031 

*037 

777 

89  042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

778 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 

779 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 

780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

781 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 

5 

782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 

783 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

1 

0.5 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

2 
3 
4 

1.0 
1.5 
2.0 

785 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

786 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

5 

2.5 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

6 

3.0 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

7 

3.5 

789 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 

8 
9 

4.0 
4.5 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

791 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 

792 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

794 

982 

988 

993 

998 

*004 

*009 

*015 

*020 

*026 

*031 

795 

90  037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

796 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 

797 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 

798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 

799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

I]    8000 — Common  Logarithms  of  Numbers — I 

3500 

17 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

soo 

90  309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

801 

363 

369 

374 

380  385 

390 

396 

401 

407 

412 

802 

417 

423 

428 !  434  439 

445 

450 

455 

461 

466 

803 

472 

477 

4821  488 

493 

499 

504 

509 

515 

520 

804 

526 

531 

536 

542 

547 

553 

558 

563 

569 

574 

805 

580 

585 

590 

596 

601 

607 

612 

617 

623 

628 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

807 

687 

693 

698 

703 

709 

714 

720 

725 

730  736 

808 

741 

747 

752 

757 

763 

768 

773 

779 

784|  789 

809 

795 

800 

806 

811 

816 

822 

827 

832 

838j  843 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891  897 

811 

902 

907 

913 

918 

924 

929 

934 

940 

945  950 

6 

812 

956 

961 

966 

972  977 

982 

988 

993 

998  *004 

813 

91  009 

014 

020 

025!  030 

036 

041 

046 

052  057 

1 

0.6 

814 

062 

068 

073 

078 

084 

089 

094 

100 

105  110 

2 
3 
4 

1.2 

1.8 
2.4 

815 

116 

121 

126 

132 

137 

142 

148 

153 

158  164 

816 

169 

.  174 

228 

180 

185 

190 

196 

201 

206 

212  217 

5 

3.0 

817 

222 

233 

238 

243 

249 

254 

259 

265  270 

6 

3.6 

818 

275 

281 

286 

291 

297 

302 

307 

312 

318  323 

7 

4.2 

819 

328 

334 

339 

344 

350 

355 

360 

365 

371!  376 

8 
9 

4.8 
5.4 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424  429 

821 

434 

440 

445 

450 

455 

461 

466 

471 

477  482 

822 

487 

492 

498 

503 

508 

514 

519 

524 

529  535 

823 

540 

545 

551 

556 

561 

566 

572 

577 

582!  587 

824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

826 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 

827 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 

828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

829 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

«955 

831 

960 

965 

971 

976 

981 

986 

991 

997 

*002*007 

5 

832 

92  012 

018 

023 

028 

033 

038 

044 

049 

054  059 

833 

065 

070 

075 

080 

085 

091 

096 

101 

106  111 

1 

0.5 

834 

117 

122 

127 

132 

137 

143 

148 

153 

158!  163 

2 
3 
4 

1.0 
1.5 
2.0 

835 

169 

174 

179 

184 

189 

195 

200 

205 

210  215 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262  267 

5 

2.5 

837 

273 

278 

283 

288 

293 

298 

304 

309 

314  319 

6 

3.0 

838 

324 

330 

335 

340  345 

350 

355 

361 

366 

371 

7 

3.5 

839 

376 

381 

387 

392|  397 

402 

407 

412 

418 

423 

8 
9 

4.0 
4.5 

840 

428 

433 

438 

443  449 

454 

459 

464 

469 

474 

841 

480 

485 

490 

495  500 

505 

511 

516 

521!  526 

842 

531 

536 

542 

547!  552 

557 

562 

567 

572  578 

843 

583 

588 

593 

598,  603 

609 

614 

619 

624  629 

844 

634 

639 

645 

650 

655 

660 

665 

670 

675  681 

845 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 

846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 

848 

840 

845 

850 

855 

860 

865 

870 

875 

8811  886 

849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988  | 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8  |  9  |   PP 

18 

8500 — Common  Logarithm 

s  of  Numbers — 

9000    [I 

N 

0 

1 

2 

3 

4 

5- 

'  6 

7 

8 

9 

PP 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

851 

993 

998  *0Q3 

*008 

*013 

*018 

*024 

*029 

*034 

*039 

852 

93  044 

049 

054 

059 

064 

069 

075 

080 

085 

090 

853 

095 

100 

105  110 

115 

120 

125 

131 

136 

141 

854 

146 

151 

156  161 

166 

171 

176 

181 

186 

192 

i 

855 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 

856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

6 

857 

298 

303 

308 

313 

318 

323 

328 

334 

339 

344 

858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

1 

0.6 

859 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

2 
3 
4 

1.2 
1.8 
2.4 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

861 

500 

505 

510 

515 

520 

526 

531 

536 

541 

546 

5 

3.0 

862 

551 

556 

561 

566 

571 

576 

581 

586 

591 

596 

6 

3.6 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

7 

4.2 

864 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 

8 
9 

4.8 

865 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 

5.4 

866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

869 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

871 

94  002 

007 

012  017 

022 

027 

032 

037 

042 

047 

5 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

873 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

1 

0.5 

874 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

2 
3 
4 

1.0 
1.5 
2.0 

875 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

5 

2.5 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

6 

3.0 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

7 

3.5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

8 
9 

4.0 
4.5 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

881 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 

882 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 

883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

886 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

4 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

888 

841 

846 

851 

856  861 

866 

871 

876 

880  885 

1 

0.4 

889 

890 

895 

900 

9051  910 

915 

919 

924 

929 

934 

2 
3 
4 

0.8 
1.2 
1.6 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

891 

988 

993 

998  *002 

*007 

*012 

*017 

*022 

*027  *032 

5 

2.0 

892 

95  036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

6 

2.4 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

7 

2.8 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

8 
9 

3.2 
3.6 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 

898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

899 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

I]    9000 — Common  Logarithms  of  Numbers — 9500 

19 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

900 

95  424 

429 

434 

439 

487 

444 

448 

453 

458 ' 

463 

468 

901 

472 

477 

482 

492 

497 

501 

506 

511 

516 

902 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 

903 

569 

574  578 

583 

588 

593 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

905 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

91© 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

911 

952  957 

961 

966 

971 

976 

980 

985 

990 

995 

5 

912_ 

999  *004  *009 

*014) 

oef 

*019 

*023 

*028 

*033> 
080 

*038  *042 

913 

96  047 

Wo»8 

052 

057 

066 

071 

118 

076 

085 

090 

1 

0.5 

914 

099 

104 

019" 

114 

123 

128 

133 

137 

2 
3 
4 

1.0 
1.5 
2.0  , 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

916 

190  194 

199 

204 

209 

213 

218 

223 

227 

232 

5 

2.5 

917 

237|  242 

246 

251 

256 

261 

265 

270 

275 

280 

6 

3.0 

918 

284!  289 

294 

298 

303 

308 

313 

317 

322 

327 

7 

3.5 

919 

332,  336 

341 

346 

350 

355 

360 

365 

369 

374 

8 
9 

4.0 
4.5 

929 

379  384 

388 

393 

398 

402 

407 

412 

417 

421 

921 

426[  431 

435 

440 

445 

450 

454 

459 

464 

468 

922 

473  478 

483 

487 

492 

497 

501 

506 

511 

515 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

927 

708  713 

717 

722 

727 

731 

736 

741 

745 

750 

928 

755  759 

764 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

931 

895 

900|  904 

909 

914 

918 

923 

928 

932 

937 

4 

932 

942 

946  951 

956 

960 

965 

970 

974 

979 

984 

933 

988 

993 
039 

997 

*002 

*007 

*011 

*016 

*021 

*025  *030 

1 

0.4 

934 

97  035 

044 

049 

053 

058 

063 

067 

072 

077 

2 
3 
4 

0.8 
1.2 
1.6 

935 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

5 

2.0 

937 

174,  179 

183 

188 

192 

197 

202 

206 

211 

216 

6 

2.4 

938 

220  225 

230 

234 

239 

.243 

248 

253 

257 

262 

7 

2.8 

939 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

8 
9 

3.2 
3.6 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

941 

359 

364 

368 

373 

377 

382 

387 

391 

396;  400 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442  447 

943 

451 

456 

460 

465 

470 

474 

479 

1  483 

488  493 

944 

497 

502 

506 

511 

516 

520 

525 

529 

534,  539 

945 

543 

548 

552 

557 

562 

566 

■  571 

575 

5801  585 

946 

589  594 

598 

603 

607 

612 

617 

621 

626!  630 

947 

635  640 

644 

649 

653 

658 

663 

i  667 

672  676 

948 

681  685 

690 

1  695 

699 

704 

708 

713 

717 

722 

949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

959 

772 

777 

782 

786 

791 

795 

800 

804 

80S 

813 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8  |  9 

PP 

20 

9500— 

Common  Logarithms  of 

Numbers — 

10000   [I 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

95© 

97  772 

777 

782 

786 
832 

791 

795 

800 

804 

809  813 

951 

818 

823 

827 

836 

841 

845 

850 

855  859 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900  905 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946  950 

954 

955 

959 

964 

968 

973 

978 

982 

987 

991  996 

955 

98  000 

005 

009 

014 

019 

023 

028 

032 

037 

041 

956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

957 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 

958 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

961 

HI 

-  277 

281 

286 

290 

295 

299 

304 

308 

313 

5 

962 

322 

327 

331 

336 

340 

345 

349 

354 

358 

963 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

1 

0.5 

964 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

2 
3 
4 

1.0 
1.5 
2.0 

965 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

966 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

5 

2.5 

967 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

6 

3.0 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

7 

3.5 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

8 
9 

4.0 
4.5 

97© 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

971 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

975 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 

976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

977 

989 

994 

998 

*003 

*007 

*012 

*016 

*021 

*025 

*029 

978 

99  034 

038 

043 

047 

052 

056 

061 

065 

069 

074 

979 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

4 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

983 

255 

260 

264 

269 

273 

277 

282 

286 

291 

.295 

1 

0.4 

984 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

2 
3 
4 

0.8 
1.2 
1.6 

985 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

5 

2.0 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

6 

2.4 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

7 

2.8 

989 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

8 
9 

3.2 
3.6 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599  603 

991 

607 

612 

616 

621 

625 

629 

634 

638 

642  647 

992 

651 

656 

660 

664 

669 

673 

677 

682 

686  691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730  734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

7741  778 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817  822 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861  865 

997 

870 

874 

878 

883 

887 

891 

896 

900 

904  909 

998 

913 

917 

922 

926 

930 

935 

939 

944 

948  952 

999 

957 

961 

965 

970 

974 

978 

983 

987 

991  996 

lOOO 

00  000 

004 

009 

013 

017 

022 

026 

030 

035  039 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

TABLE  II 

Common  Logarithms  of  the  Trigonometric  Functions 

For  each  second  from  0'  to  3'  and  from  89°  57 '  to  90° 

For  every  ten  seconds  from  3'  to  2°  and  from  88°  to  89°  57' 

For  each  minute  from  2°  to  88° 

to  Five  Decimal  Places 


II] 

0'- 

-Logarithms  of  Trigonometric  Functions— 

-3' 

23 

L  Sin  and  L  Tan 

L  Sin  and  L  Tan 

// 

0' 

V 

2' 

// 

n 

0' 

V 

2' 

// 

o 

1 

2 
3 
4 

4.68  557 
4.98  660 
5.16  270 
5.28  763 

6.46  373 

6.47  090 

6.47  797 

6.48  492 

6.49  175 

6.76  476 

6.76  836 

6.77  193 
6.77  548 
6.77  900 

60 

59 
58 
57 
56 

80 

31 
32 
33 
34 

6.16  270 

6.17  694 

6.19  072 

6.20  409 

6.21  705 

6.63  982 

6.64  462 

6.64  936 

6.65  406 
6.65  870 

6.86  167 
6.86  455 

6.86  742 

6.87  027 
6.87  310 

30 

29 
28 
27 
26 

5 

6 

7 
8 
9 

5.38  454 
5.46  373 
5.53  067 
5.58  866 
5.63  982 

6.49  849 

6.50  512 

6.51  165 

6.51  808 

6.52  442 

6.78  248 
6.78  595 

6.78  938 

6.79  278 
6.79  616 

55 

54 
53 
52 
51 

35 

36 
37 
38 
39 

6.22  964 

6.24  188 

6.25  378 

6.26  536 

6.27  664 

6.66  330 

6.66  785 

6.67  235 

6.67  680 

6.68  121 

6.87  591 

6.87  870 

6.88  147 
6.88  423 
6.88  697 

25 

24 
23 
22 
21 

io 

11 
12 
13 

14 

5.68  557 
5.72  697 
5.76  476 
5.79  952 
5.83  170 

6.53  067 

6.53  683 

6.54  291 

6.54  890 

6.55  481 

6.79  952 

6.80  285 
6.80  615 

6.80  943 

6.81  268 

50 

49 

48 
47 
46 

40 

41 
42 
43 
44 

6.28  763 

6.29  836 

6.30  882 

6.31  904 

6.32  903 

6.68  557 

6.68  990 

6.69  418 

6.69  841 

6.70  261 

6.88  969 

6.89  240 
6.89  509 

6.89  776 

6.90  042 

20 

19 
18 
17 
16 

15 

16 
17 
18 
19 

5.86  167 
5.88  969 
5.91  602 
5.94  085 
5.96  433 

6.56  064 

6.56  639 

6.57  207 

6.57  767 

6.58  320 

6.81  591 

6.81  911 

6.82  230 
6.82  545 
6.82  859 

45 

44 
43 

42 
41 

45 

46 
47 
48 
49 

6.33  879 

6.34  833 

6.35  767 

6.36  682 

6.37  577 

6.70  676 

6.71  088 
6.71  496 

6.71  900 

6.72  300 

6.90  306 
6.90  568 

6.90  829 

6.91  088 
6.91  346 

15 

14 
13 
12 
11 

20 

21 
22 
23 
24 

5.98  660 
6.00  779 
6.02  800 
6.04  730 
6.06  579 

6.58  866 

6.59  406 

6.59  939 

6.60  465 
6.60  985 

6.83  170 
6.83  479 

6.83  786 

6.84  091 
6.84  394 

40 

39 
38 
37 
36 

50 

51 
52 
53 
54 

6.38  454 

6.39  315 

6.40  158 

6.40  985 

6.41  797 

6.72  697 

6.73  090 
6.73  479 

6.73  865 

6.74  248 

6.91  602 

6.91  857 

6.92  110 
6.92  362 
6.92  612 

IO 

9 

8 
7 
6 

25 

26 
27 

28 
29 

6.08  351 

6.10  055 

6.11  694 

6.13  273 

6.14  797 

6.61  499 

6.62  007 

6.62  509 

6.63  006 
6.63  496 

6.84  694 

6.84  993 

6.85  289 
6.85  584 
6.85  876 

35 

34 
33 
32 
31 

55 

56 
57 
58 
59 

6.42  594 

6.43  376 

6.44  145 

6.44  900 

6.45  643 

6.74  627 

6.75  003 
6.75  376 

6.75  746 

6.76  112 

6.92  861 

6.93  109 
6.93  355 
6.93  599 
6.93  843 

5 

4 
3 

2 

1 

30 

6.16  270 

6.63  982 

6.86  167 

30 

OO 

6.46  373 

6.76  476 

6.94  085 

O 

rr 

59' 

58' 

57' 

// 

// 

59' 

58' 

57' 

// 

L  Cos  and  L  Cot 

L  Cos  and  L  Cot 

89057, — Logarithms  of  Trigonometric  Functions — 90c 

— 10  should  be  written  after  every  logarithm  taken  from  this  page. 


24 

3'— 

Logarithms  of 

Trigonometric  Functions- 

-20' 

[II 

/    // 

LSin 

LCos 

LTan 

//  / 

10    0 

7.46  373 

0.00  0007.46  373 

0  50 

10 

7.47  090 

0.00  0007.47  091 

50 

20 

7.47  797 

0.00  0007.47  797 

40 

30 

7.48  491 

0.00  0007.48  492 

30 

Logarithms  of  functions  of  angle 

sless 

40 

7.49  175 

0.00  0007.49  176 

20 

than  3'  or  greater  than  89°57'  are  found 

50 

7.49  8490. 

10 

on  the  preceding  page. 

11    0 

7.50  5120.00 

0  49 

10 

7.51   1650.00  0007.51   165 

50 

20 

7.51  808 

0.00  0007.51  809 

40 

30 

7.52  442 

0.00  0007.52  443 

30 

40 

7.53  067 

0.00  0007.53  067 

20 

50 

7.53  683 

0.00  000 

7.53  683 

10 

12     0 

7.54  291 

0.00  0007.54  291 

0  48 

10 

7.54  8900.00  0007.54  890 

50 

20 
30 

7.55  481 

7.56  064 

0.00  0007.55  481  40 

0.00  000 

7.56  064  30 

/   // 

LSin 

LCos 

LTan 

//   / 

40 

7.56  639 

0.00  000 

7.56  639  20 

50 
13    0 

7.57  206 
7.57  767 

0.00  000 
0.00  000 

7.57  207 
7.57  767 

10 
0  47 

3    0 

6.94  085 

0.00  000 

6.94  085 

0  57 

10 

6.96  433 

0.00  000  6.96  433 

50 

10 

7.58  3200.00  000 

7.58  320 

50 

20 

6.98  660  0.00  000 

6.98  661 

40 

20 

7.58  866  0.00  000 

7.58  867 

40 

30 

7.00  779  0.00  000 

7.00  779 

30 

30 

7.59  4060.00  000 

7.59  406 

30 

40 

7.02  800  0.00  000 

7.02  800 

20 

40 

7.59  939 

0.00  000 

7.59  939 

20 

50 

7.04  730  0.00  000 

7.04  730 

10 

50 

7.60  465 

0.00  000 

7.60  466 

10 

4    0 

7.06  579  0.00  000 

7.06  579 

0  56 

14    0 

7.60  985 

0.00  000 

7.60  986 

0  46 

10 

7.08  3510.00  000  7.08  352 

50 

10 

7.61  499 

0.00  000)7.61  500 

50 

20 

7.10  055l0.00  000 

7.10  055 

40 

20 

7.62  007 

0.00  000  7.62  008 

40 

30 

7.11  694  0.00  000 

7.11  694 

30 

30 

7.62  509 

0.00  0007.62  510 

30 

40 

7.13  273  0.00  000 

7.13  273 

20 

40 

7.63  006 

0.00  000  7.63  006 

20 

50 

7.14  797  0.00  000 

7.14  797 

10 

50 

7.63  496 

0.00  0007.63  497 

10 

5    0 

7.16  27o'o.OO  000 

7.16  270 

0  55 

15     0 

7.62  982 

0.00  0007.63  982 

0  45 

10 

7.17  694  0.00  000  7.17  694 

50 

10 

7.64  461 

0.00  0007.64  462 

50 

20 

7.19  0720.00  000 

7.19  073 

40 

20   7.64  936  0.00  0007.64  937 

40 

30 

7.20  409j 0.00  000 

7.20  409 

30 

30    7.65  406|0.00  0007.65  406 

30 

40 

7.21  705  0.00  000 

7.21  705 

20 

40  |7.65  870  0.00  00017.65  871 

20 

50 

7.22  964(0.00  000 

7.22  964 

10 

50 

7.66  330 

0.00  0007.66  330 

10 

6    0 

7.24  188  0.00  000  7.24  188 

0  54 

16    0 

7.66  784 

0.00  0007.66  785 

0  44 

10 

7.25  378  0.00  000 

7.25  378 

50 

10 

7.67  235 

0.00  000  7.67  235 

50 

20 

7.26  5360.00  000 

7.26  536 

40 

20 

7.67  680 

0.00  0007.67  680 

40 

30 

7.27  664  0.00  000 

7.28  763  0.00  000 

7.27  664 

30 

30 

7.68  121 

0.00  0007.68  121 

30 

40 

7.28  764 

20 

40 

7.68  657 

9.99  999 

7.68  558 

20 

50 

7.29  836  0.00  000,7.29  836 

10 

50 

7.68  989 

9.99  999 

7.68  990 

10 

7    0 

7.30  882  0.00  0007.30  882 

0  53 

17    0 

7.69  417 

9.99  999 

7.69  418 

0  43 

10 

7.31  904  0.00  000 

7.31  904 

50 

10 

7.69  841|9.99  999 

7.69  842 

50 

20 

7.32  90310.00  000 

7.32  903 

50 

20 

7.70  2619.99  999 

7.70  261 

40 

30 

7.33  879|0.00  000 

7.33  879 

30 

30 

7.70  6769.99  9997.70  677 

30 

40 

7.34  833 
t.35  767 

0.00  000 

7.34  833 

20 

40 

7.71  0889.99  999 

7.71  088 

20 

50 

0.00  000 

7.35  767 

10 

50 

7.71  496,9.99  999 

7.71  496 

10 

8    0 

7.36  682 

0.00  000  7.36  682 

0  52 

18    0 

7.71  900i9.99  9997.71  900 

0  42 

10 

7.37  577  0.00  000  7.37  577 

50 

10 

7.72  3009.99  999  7.72  301!  50 

20 

7.38  454  0.00  000  7.38  455 

40 

20 

7.72  697  9.99  9997.72  697  40 
7 .  73  090  9 .  99  999  7 .  73  090  30 

30 

7.39  3140.00  000 

7.39  315 

30 

30 

40 

7.40  158  0.00  000 

7.40  158 

20 

40 

7.73  4799.99  9997.73  480  20 

50 

7.40  985  0.00  000 

7.40  985 

10 

50 

7.73  8659.99  999 

7.73  866 

10 

9    0 

7.41  797 

0.00  000 

7.41  797 

0  51 

19    0 

7.74  248 

9.99  999 

7.74  248 

0  41 

10 

7.42  594 

0.00  000  7.42  594 

50 

10 

7.74  627 

9.99  999 

7.74  628 

50 

20 

7.43  3760.00  0007.43  376 

40 

20 

7.75  003 

9.99  999 

7.75  004 

40 

30 

7.44  145  0.00  000  7.44  145 

30 

30 

7.75  376 

9.99  999 

7.75  377 

30 

40 

7.44  9000.00  000 

7.44  900 

20 

40 

7.75  745 

9.99  999 

7.75  746 

20 

50 

7.45  643  0.00  000 

7.45  643 

10 

50 

7.76  112 

9.99  999 

7.76  113 

10 

10    0 

7.46  373 

0.00  000 

7.46  373 

0  50 

20    0 

7.76  475 

9.99  999 

7.76  476 

0  40 

/    // 

LCos 

LSin 

LCot 

//    / 

/    // 

LCos 

LSin 

LCot 

//    / 

89°40'—  Logarithms  of  Trigonometric  Functions— 89°57' 

— 10  should  be  written  after  every  logarithm  taken  from  this  page,  except  those 
Wat  are  0.00000 


II] 

20'- 

-Logarithms  of  Trigonometric  Functions — 40' 

25 

1  If 

LSin 

LCos 

LTan 

//  / 

/  // 

LSin 

L  Cos 

LTan 

//  / 

20  0 

7.76  475 

9.99  999 

7.76  476 

0  40 

30  0 

7.94  084 

9.99  998 

7.94  086 

0  30 

10 

7.76  836  9.99  99917.76  837 

50 

10 

7.94  3259.99  998 

7.94  326 

50 

20 

7.77  193  9.99  999  7.77  194|  40 

20 

7.94  5649.99  99817.94  566 

40 

30 

7.77  548  9.99  999  7.77  549|  30 

30 

7.94  8029.99  99817.94  804  30 

40 

7.77  899  9.99  999 

7.77  900  20 

40 

7.95  039  9.99  998  7.95  040  20 

50 

7.78  248j9.99  999 

7.78  249 

10 

50 

7.95  274 

9.99  998 

7.95  276  10 

21  0 

7.78  594  9.99  999 

7.78  595 

0  39 

31  0 

7.95  508 

9.99  998 

7.95  510  0  29 

10 

7.78  938  9.99  999 

7.78  938 

50 

10 

7.95  741 

9.99  998  7.95  743  50 

20 

7.79  278  9.99  999 

7.79  279 

40 

20 

7.95  973  9.99  998:7.95  974|  40 

30 

7.79  616  9.99  99917.79  617 

30 

30 

7.96  203  9.99  9987.96  205  30 

40 

7 .  79  952  9 .  99  999 

7.79  952 

20 

40 

7.96  432  9.99  998 

7.96  434  20 

50 

7.80  284:9.99  999 

7.80  285 

10 

50 

7.96  660:9.99  998 

7.96  662  10 

22  0 

7.80  615'9.99  999 

7.80  615 

0  38 

32  0 

7.96  887  9.99  998 

7.96  889  0  28 

10 

7.80  942  9.99  999 

7.80  943 

50 

10 

7.97  113  9.99  998  7.97  114  50 

20 

7.81  268  9.99  999 

7.81  269 

40 

20 

7.97  337  9.99  998  7.97  339!  40 

30 

7.81  5919.99  999  7.81  591 

30 

30 

7.97  560!9.99  998 

7.97  562i  30 

40 

7.81  911  9.99  999 

7.81  912 

20 

40 

7.97  782  9.99  998 

7.97  784  20 

50 

7.82  229,9.99  999 

7.82  230 

10 

50 

7.98  003  9.99  998 

7 .  98  005 

10 

23  0 

7.82  545  9.99  999 

7.82  546 

0  37 

33  0 

7.98  223  9.99  998 

7.98  225 

0  27 

10 

7.82  859  9.99  999 

7.82  860 

50 

10 

7.98  4429.99  998 

7.98  444 

50 

20 

7.83  170  9.99  999 

7.83  171 

40 

20 

7.98  660  9.99  998  7.98  662 

40 

30 

7.83  479  9.99  999 

7.83  480 

30 

30 

7.98  876  9.99  998j7.98  878  30 

40 

7.83  786i9.99  999 

7.83  787 

20 

40 

7.99  092 

9.99  998 

7.99  094 

20 

50 

7.84  091 

9.99  999 

7.84  092 

10 

50 

7.99  306 

9.99  998 

7.99  308 

10 

24  0 

7.84  393 

9.99  999 

7.84  394 

0  36 

34  0 

7.99  520 

9.99  998 

7.99  522 

0  26 

10 

7.84  694 

9.99  999 

7.84  695  50 

10 

7.99  732 

9.99  998 

7.99  734 

50 

20 

7.84  992 

9.99  999 

7 . 84  994  40 

20 

7.99  943 

9.99  998 

7.99  946 

40 

30 

7.85  289i9.99  999 

7 .  85  290!  30 

30 

8.00  154 

9.99  998 

8.00  156 

30 

40 

7.85  583 

9.99  999 

7.85  584 

20 

40 

8.00  363 

9.99  998 

8.00  365 

20 

50 

7.85  876 

9.99  999 

7.85  877 

10 

50 

8.00  571 

9.99  998 

8.00  574 

10 

25  0 

7.86  166 

9.99  999 

7.86  167 

0  35 

35  0 

8.00  779 

9.99  998 

8.00  781 

0  25 

10 

7.86  455 

9.99  999 

7.86  456 

50 

10 

8.00  985  9.99  998  8.00  987 

50 

20 

7.86  741 

9.99  999  7.86  743 

40 

20 

8.01  1909.99  9988.01  193 

40 

30 

7.87  026 

9.99  999  7.87  027  30 

30 

8.01  395  9.99  998  8.01  397 

30 

40 

7.87  309 

9.99  999  7.87  310  20 

40 

8.01  598  9.99  998  8.01  600 

20 

50 

7.87  590 

9.99  999 

7.87  591 

10 

50 

8.01  801  9.99  998j8.01  803 

10 

26  0 

7.87  870 

9.99  999 

7.87  871 

0  34 

36  0 

8.02  OO2I9.99  998  8.02  004 

0  24 

10 

7.88  147 

9.99  999 

7.88  148 

50 

10  8.02  203  9.99  998  8.02  205 

50 

20  7 . 88  423 

9 .  99  999  7 .  88  424 

40 

20|8.02  402  9.99  998  8.02  405 

40 

30 

7.88  6979.99  999  7.88  698 

30 

.30|8.02  601|9.99  9988.02  604 

30 

40 

7.88  969 

9.99  999  7.88  970  20 

40  8.02  799 

9.99  998 

8.02  801 

20 

50 

7.89  240 

9.99  999  7.89  241 

10 

50 

8.02  996 

9.99  998 

8.02  998 

10 

27  0 

7.89  509 

9.99  999  7.89  510 

0  33 

37  0 

8.03  192 

9.99  997 

8.03  194 

0  23 

10 

7.89  776 

9.99  99917.89  777 

50 

10 

8.03  387 

9.99  997 

8.03  390 

50 

20 

7.90  041 

9.99  999  7.90  043 

40 

20 

8.03  581 

9.99  997  8.03  584 

40 

30 

7.90  305 

9.99  999 

7.90  307 

30 

30 

8.03  775 

9.99  997|8.03  777 

30 

40 

7.90  568 

9.99  999 

7.90  569 

20 

40 

8.03  967 

9.99  99718.03  970 

20 

50 

7.90  829 

9.99  999 

7.90  830 

10 

50 

8.04  159 

9.99  997  8.04  162 

10 

28  0 

7.91  088 

9.99  999 

7.91  089 

0  32 

38  0 

8.04  350 

9.99  997  8.04  353 

0  22 

10 

7.91  346 

9.99  999 

7.91  347 

50 

•   10 

8.04  540 

9.99  997  8.04  543 

50 

20  7.91  602 

9.99  999 

7.91  603 

40 

20 

8.04  729  9.99  997  8.04  732 

40 

30 

7.91  857 

9.99  999 

7.91  858 

30 

30 

8.04  918  9.99  997  8.04  921 

30 

40 

7.92  110 

9.99  998 

7.92  111 

20 

40 

8.05  105  9.99  997  8.05  108 

20 

50 

7.92  362 

9.99  998 

7.92  363 

10 

50 

8.05  292 

9.99  997 

8.05  295 

10 

29  0 

7.92  612 

9.99  998 

7.92  613 

0  31 

39  0 

8.05  478 

9.99  997 

8.05  481 

0  21 

10 

7.92  861 

9.99  998 

7.92  862 

50 

10 

8.05  663 

9.99  997 

8.05  666 

50 

20 

7.93  108 

9.99  998 

7.93  110 

40 

20 

8.05  848 

9.99  997  8.05  851 

40 

30 

7.93  354 

9.99  998 

7.93  356 

30 

30 

8.06  031 

9.99  997  8.06  034 

30 

40 

7.93  599 

9.99  998 

7.93  601 

20 

40 

8.06  214 

9.99  9978.06  217 

20 

50 

7.93  842 

9.99  998 

7.93  844 

10 

50 

8.06  396 

9.99  997 

8.06  399 

10 

30  0 

7.94  084 

9.99  998 

7.94  086 

0  30 

40  0 

8.06  578 

9.99  997 

8.06  581 

0  20 

/  // 

LCos 

LSin 

LCot 

//  / 

/  // 

LCos 

LSin 

LCot 

//  / 

89°20'— Logarithms  of  Trigonometric  Functions—  89°40' 

- 10  should  be  written  after  every  logarithm  taken  from  this  page. 


26 

40'- 

-Logarithms  of  Trigonometric  Functions — 1° 

[II 

/.  // 

LSin 

LCos 

LTan 

//   / 

/   // 

LSin 

1 

L  Cos  I L  Tan 

//   / 

40     0 

8.  OS  57? 

9.99  997 

8.06  581 

0  20 

50    0 

8.16  268 

9.99  995  8.16  273 

0  10 

10 

8.06  758  9.99  997  8.06  761  50 

10 

8.16  413 

9.99  995  8.16  417 

50 

20 

8.06  9389.99  997  8.06  941  40 

20 

8.16  557  9.99  995  8.16  561 

40 

30 

8.07   117,9.99  997  8.07   120  30 

30 

8.16  700  9.99  995  8.16  705 

30 

40 

8.07  29£ 

9.99  997  8.07  299  20 

40 

8.16  84319.99  995  8.16  848 

20 

50 

8.07  472 

9.99  997  8.07  476  10 

50 

8.16  986  9.99  995  8.16  991 

10 

41    0 

8.07  65C 

9.99  997  8.07  653     0  19 

51    0 

8.17  128  9.99  995  8.17  133 

0     9 

10 

8.07  826 

9.99  997(8.07  829  50 

10 

8.17  270  9.99  995(8.17  275 

50 

20 

8.08  002 

9.99  997  8.08  005  40 

20 

8.17  4119.99  9958.17  416 

40 

30 

8.08  176 

9.99  997  8.08  180  30 

30 

8.17  552  9.99  995 

8.17  557 

30 

40 

8.08  35019.99  997  8.08  354  20 

40 

8.17  692 

9.99  995 

8.17  697 

20 

50 

8.08  524 

9.99  997  8.08  527  10 

50 

8.17  832 

9.99  995 

8.17  837 

10 

42     0 

8.08  696 

9.99  997  8.08  700     0  18 

52     0 

8.17  971 

9.99  995 

8.17  976 

0    8 

10 

8.08  868  9.99  997  8.08  872  50 

10 

8.18  110 

9.99  995 

8.18  115 

50 

20 

8.09  040  9.99  997  8.09  043  40 

20 

8.18  249 

9.99  995 

8.18  254 

40 

30 

8.09  210  9.99  997  8.09  214  30 

30 

8.18  387 

9.99  995  8.18  392 

30 

40 

8.09  380  9.99  997  8.09  384  20 

40 

8.18  524 

9.99  995 

8.18  530 

20 

50 

8.09  550|9.99  997  8.09  553j  10 

50 

8.18  662 

9.99  995 

8.18  667 

10 

43    0 

8.09  718  9.99  99718.09  722     0  17 

53     0 

8.18  798 

9.99  995 

8.18  804 

0    7 

10 

8.09  886  9.99  9978.09  89/) 

50 

10 

8.18  935 

9.99  995 

8.18  940 

50 

20 

8.10  054  9.99  997  8.10  057 

40 

20 

8.19  071 

9.99  995J8.19  076 

40 

30 

8.10  220  9.99  997  8.10  224 

30 

30 

8.19  206 

9.99  995  8.19  212 

30 

40 

8.10  386  9.99  997  8.10  390 

20 

40 

8.19  341 

9.99  995  8.19  347 

20 

50 

8.10  552  9.99  996  8.10  555 

10 

50 

8.19  476 

9.99  995  8.19  481 

10 

44     0 

8.10  717  9.99  996  8.10  720 

0  16 

54    0 

8.19  610 

9.99  995  8.19  616 

0     6 

10 

8.10  881;9.99  996  8.10  884 

50 

10 

8.19  744 

9.99  995  8.19  749 

50 

20 

8.11  044  9.99  996  8.11  048 

40 

20 

8.19  877  9.99  995|8.19  883 

40 

30 

8.11  207!9.99  9968.11  211 

30 

30 

8.20  010  9.99  995  8.20  016 

30 

40 

8.11  370  9.99  996  8.11  373 

20 

40 

8.20  143 

9.99  995  8.20  149 

20 

50 

8.11  531 

9.99  996 

8.11  535 

10 

50 

8.20  275 

9.99  994 

8.20  281 

10 

45     0 

8.11  693 

9.99  996 

8.11  696 

0  15 

55    0 

8.20  407 

9.99  994 

8.20  413 

0     5 

10 

8.11   853  9.99  996  8.11  857 

50 

10 

8.20  538 

9.99  994 

8.20  544 

50 

20 

8.12  0139.99  996:8.12  017 

40 

20 

8.20  669 

9.99  994 

8.20  675 

40 

30 

8.12  1729.99  996  8.12   176 

30 

30 

8.20  800 

9.99  994 

8.20  806 

30 

40 

8.12  331 

9.99  996  8.12  335 

20 

40 

8.20  930 

9.99  994 

8.20  936 

20 

50 

8.12  489 

9.99  996 

8.12  493 

10 

50 

8.21  060 

9.99  994 

8.21  066 

10 

46     0 

8.12  647 

9.99  996 

8.12  651 

0  14 

56    0 

8.21   189 

9.99  994 

8.21   195 

0     4 

10 

8.12  804 

9.99  996 

8.12  808 

50 

10 

8.21  319 

9.99  994 

8.21  324 

50 

20 

8.12  961 

9.99  996 

8.12  965 

40 

20 

8.21  447 

9.99  994 

8.21  453 

40 

30 

8.13   117 

9.99  996 

8.13  121 

30 

30 

8.21   576 

9.99  994 

8.21  581 

30 

40 

8.13  272 

9.99  996 

8.13  276 

20 

40 

8.21  703 

9.99  994 

8.21  709 

20 

50 

8.13  427 

9.99  996 

8.13  431 

10 

50 

8.21  831 

9.99  994 

8.21  837 

10 

47    0 

8.13  581 

9.99  996 

8.13  585 

0  13 

57    0 

8.21  958 

9.99  994 

8.21  964 

0    3 

10 

8.13  735 

9.99  996  8.13  739 

50 

10 

8.22  085 

9.99  994 

8.22  091 

50 

20 

8.13  888 

9.99  996  8.13  892 

40 

20 

8.22  211 

9.99  994 

8.22  217 

40 

30 

8.14  041 

9.99  996 

8.14  045 

30 

30 

8.22  337 

9.99  994 

8.22  343 

30 

40 

8.14  193 

9.99  996 

8.14  197 

20 

40 

8.22  463 

9.99  994 

8.22  469 

20 

50 

8.14  344 

9.99  996 

8.14  348 

10 

50 

8.22  588 

9.99  994 

8.22  595 

10 

48    0 

8.14  495 

9.99  996 

8.14  500 

0  12 

58    0 

8.22  713 

9.99  994 

8.22  720 

0    2 

10 

8.14  646 

9.99  99* i 

8.14  650 

50 

10 

8.22  838 

9.99  994 

8.22  844 

50 

20 

8.14  796 

9.99  996  8.14  800 

40 

20 

8.22  962 

9.99  994 

8.22  96S    40 

30 

8.14  945  9.99  996,8.14  950 

30 

30 

8.23  086 

9.99  994 

8.23  092 

30 

40 

8.15  094  9.99  996  8.15  099 

20 

40 

8.23  210 

9.99  994 

8.23  216 

20 

50 

8.15  243 

9.99  996 

8.15  247 

10 

50 

8.23  333 

9.99  994 

8.23  339 

10 

49    0 

8.15  391 

9.99  996 

8.15  395 

0  11 

59     0 

8.23  456 

9.99  994 

8.23  462 

0     1 

10 

8.15  538 

9.99  996 

8.15  543 

50 

10 

S.23  578 

9.99  994 

8.23  585 

50 

20 

8.15  685  9.99  996 

8.15  690 

40 

20 

S.23  700 

9.99  9<>4 

8.23  707    40 

30 

8.15  832 

9.99  996  8.15  836 

30 

30  18.23  822 

9.99  993  8.23  829    30 

40 

8.15  978 

9.99  995 

8.15  982 

20 

40 

S.23  944 

9.99  993  8.23  950 

20 

50 

8.16  123 

9.99  995 

8.16  128  10 

50 

S.24  065 

9.99  993 

8.24  071 

10 

60    o 

8.16  268 

9.99  995 

8.16  273     0  10 

60    0 

S.24   186 

9.99  993 

8.24  192 

0     0 

/     // 

LCos 

LSin 

L  Cot     "  ' 

/    // 

LCos 

LSin 

LCot 

u    f 

89°— Logarithms  of  Trigon 

ometric  Functions— 89°20' 

-10sh 

ould  be  w 

ritten  aft( 

jr  ever j 

r  logaril 

,hm  taken 

from  this 

5  page. 

II] 

1° — Logarithms  of  Trigonometric  Functions— 

-1°20' 

27 

/   // 

LSin 

LCos 

LTan 

//  / 

/    // 

LSin 

LCos 

LTan 

//  / 

0    0 

8.24  186 

9.99  993 

8.24  192 

0  60 

10    0 

8.30  879 

9.99  991 

8.30  888 

0  50 

10  8.24  306  9.99  993  8.24  313 

50 

10 

8.30  983  9.99  991 

8.30  992(  50 

20 

8.24  426 

9.99  993  8.24  433 

40 

20  8.31  0869.99  991:8.31  095j  40 

30 

8.24  546 

9.99  993  8.24  553 

30 

30 

8.31   188  9.99  9918.31   198   30 

40 

8.24  665 

9.99  9938.24  672 

20 

40 

8.31  2919.99  9918.31  300   20 

50 

8.24  785  9.99  993  8.24  791 

10 

50 

8.31  393  9.99  991J8.31  403 

10 

1    0 

8.24  903  9.99  993  8.24  910 

0  59 

11     0 

8.31  4959.99  99118.31  505 

0  49 

10 

8.25  022  9.99  993  8.25  029 

50 

10  8.31   597  9.99  991  8.31  606 

50 

20 

8.25  1409.99  993  8.25   147 

40 

.      20  8.31  699  9.99  9918.31  708 

40 

30 

8.25  258  9.99  993  8.25  265 

30 

30  8.31  800  9.99  9918.31  809 

30 

40 

8.25  375  9.99  993J8.25  382 

20 

40  8.31  9019.99  9918.31  911 

20 

50 

8.25  493.9.99  993  8.25  500 

10 

50  8.32  0029.99  9918.32  012 

10 

2    0 

8.25  609  9.99  9938.25  616 

0  58 

12     0  8.32   103  9.99  9908.32  112 

0  48 

10 

8.25  7269.99  993  8.25  733 

50 

10  8.32  203  9.99  9908.32  213    50 

20 

8.25  84219.99  993  8.25  849 

40 

20  8.32  303  9.99  990  8.32  313   40 

30 

8.25  9589.99  9938.25  965  30 

30  8.32  403  9.99  990  8.32  413   30 

40 

8.26  074:9.99  993j8.26  081  20 

40  8.32  503  9.99  990  8.32  513    20 

50 

8.26  189  8.99  993.8.26  196 

10 

50,8.32  602  9.99  990  8.32  612 

10      1 

3    0 

8.26  304  9.99  993  8.26  312 

0  57 

13    0  8.32  702  9.99  990  8.32  711 

0  47 

10 

8.26  419  9.99  993  8.26  426 

50 

10  8.32  8019.99  990  8.32  811 

50 

20  8.26  5339.99  993  8.26  541 

40 

20(8.32  899  9.99  990  8.32  909   40 

30  8.26  648  9.99  993  8.26  655 

30 

30  8.32  998  9.99  990  8.33  008:  30 

40 

8.26  7619.99  993  8.26  769 

20 

40  8.33  096  9.99  9908.33   106    20 

50 

8.26  875 

9.99  993 

8.26  882 

10 

50 

8.33   195 

9.99  990  8.33  205    10 

4    0 

8.26  988 

9.99  992 

8.26  996 

0  56 

14    0 

8.33  292 

9.99  990'8.33  302     0  46 

10 

8.27   101  9.99  992 

8.27   109 

50 

10  8.33  390(9.99  990  8.33  400|  50 

20 

8.27  214  9.99  99218.27  221 

40 

20  8.33  488  9.99  9908.33  498   40 

30 

8.27  326  9.99  992  8.27  334  30 

30  8.33  5859.99  990  8.33  595    30 

40 

8.27  438,9.99  992  8.27  446!  20 

40  8.33  682  9.99  9908.33  692 

20 

50 

8.27  550 

9.99  992  8.27  558 

10 

50  8.33  779  9.99  990,8.33  789 

10 

5    0 

8.27  661 

9.99  992  8.27  669 

0  55 

15    0  8.33  87519.99  990  8.33  886 

0  45 

10 

8.27  773 

9.99  992(8.27  780 

50 

10  18.33  972(9.99  990  8.33  982 

50 

20 

8.27  883(9.99  992|8.27  891 !  40 

20  8.34  068  9.99  990  8.34  078 

40 

30 

8.27  994 

9.99  992(8.28  002|  30 

30  18.34  164(9.99  990(8.34  174 

30 

40 

8.28  104 

9.99  992 

8.28  112  20 

40  |8.34  260(9.99  989(8.34  270    20 

50 

8.28  215 

9.99  992 

8.28  223  10 

50 

8.34  355 

9.99  989i8.34  366 

10 

6    0 

8.28  324 

9.99  992 

8.28  332|     0  54 

16    0 

8.34  450 

9.99  989'8.34  461 

0  44 

10 

8.28  434 

9.99  992 

8.28  4421  50 

10 

8.34  546 

9.99  989  8.34  556 

50 

20 

8.28  5439.99  992  8.28  551140 

20 

8.34  640 

9.99  989  8.34  651 

40 

30 

8.28  65219.99  9928.28  660  30 

30 

8.34  735 

9.99  989  8.34  746 

30 

40 

8.28  761  9. 99  992  8.28  769 

20 

40 

8.34  830  9.99  989  8.34  840 

20 

50 

8.28  869i9.99  992  8.28  877 

10 

50 

8.34  924  9.99  989j8.34  935 

10 

7    0 

8.28  977  9.99  992  8.28  986 

0  53 

17    0 

8.35  018  9.99  989  8.35  029 

0  43 

10 

8.29  085  9.99  992(8.29  094 

50 

10 

8.35   112J9.99  9898.35  123 

50 

20 

8.29  193  9.99  992(8.29  201  40 

20 

8.35  206  9.99  989  8.35  217 

40 

30 

8.29  300  9.99  992  8.29  309 

30 

30 

8.35  299  9.99  989  8.35  310 

30 

40 

8.29  407  9.99  992  8.29  416 

20 

40 

8.35  392  9.99  989 

8.35  403 

20 

50 

8.29  514  9.99  992 

8.29  523 

10 

50 

8.35  485 

9.99  989 

8.35  497 

10 

8     0 

8.29  621  9.99  992 

8.29  629 

0  52 

18     0 

8.35  578 

9.99  989 

8.35  590 

0  42 

10 

8.29  727  9.99  991 

8.29  736 

50 

10 

8.35  671 

9.99  989 

8.35  682 

50 

20  8.29  833  9.99  9918.29  842i  40 

20 

8.35  764 

9.99  989  8.35  775 

40 

30  18.29  939  9.99  9918.29  947 

30 

30 

8.35  85619.99  989  8.35  867 

30 

40i8.30  044  9.99  9918.30  053 

20 

40 

8.35  948 

9.99  989  8.35  959 

20 

50 

8.30  150|9.99  991j8.30   158 

10 

50 

8.36  040 

9v99  989 

8.36  051 

10 

9    0 

8.30  255  9.99  991  8.30  263 

0  51 

19    0 

8.36   131 

9.99  989 

8.36  143 

0  41 

10 

8.30  359  9.99  9918.30  368 

50 

10 

8.36  223 

9.99  988  8.36  235 

50 

20 

8.30  464  9.99  9918.30  473  40 

20 

8.36  314 

9.99  988(8.36  326 

40 

30 

8.30  568  9.99  9918.30  577  30 

30 

8.36  405 

9.99  98818.36  417 

30 

40 

8.30  6729.99  991  8.30  681   20 

40 

8.36  496 

9.99  988  8.36  508 

20 

50 

8.30  776  9.99  991  j 8. 30  785 

10 

50 

8.36  587 

9.99  988 

8.36  599 

10 

10    0 

8.30  879 

9.99  991 

8.30  888 

0  50 

20    0 

8.36  678 

9.99  988 

8.36  689 

0  40 

/    // 

LCos 

LSin 

LCot 

//    / 

/    // 

LCos 

LSin 

LCot 

//    / 

88°40'— Logarithms  of  Trigonometric  Functions— 88°60/ 

— 10  should  be  written  after  every  logarithm  taken  from  this  page. 


28 

1°20'- 

-Logarithms  of 

Trigonometric  Functions 

— 1°40' 

[II 

t  // 

LSin 

LCos 

LTan 

//  f 

/    // 

LSin 

LCos 

LTan 

//  / 

20    0 

8.36  678 

9.99  988 

8.36  689 

0  40 

30     0 

8.41  792 

9.99  985 

8.41  807 

0  30 

10  18.36  7689.99  9888.36  780 

50 

10 

8.41   87219.99  985 

8.41  887 

50 

20 

8.30  S58  9.99  988  8.36  870 

40 

20 

8.41  952!9.99  985 

8.41  967 

40 

30 

8 .  36  948  9 .  99  988  8 .  36  960 

30 

30 

8.42  032  9.99  985 

8.42  048 

30 

40 

8.37  038  9.99  988  8.37  050 

20 

40 

8.42  112  9.99  985 

8.42  127 

20 

50 

8.37  128,9.99  988,8.37  140 

10 

50 

8.42   192 

9.99  985 

8.42  207 

10 

21     0 

8.37  217  9.99  98818.37  229 

0  39 

31    0 

8.42  272 

9.99  985 

8.42  287 

0  29 

10  8.37  306  9.99  988  8.37  318 

50 

10  8.42  351 

9.99  985  8.42  366 

50 

20  8.37  395  9.99  988  8.37  408 

40 

20  8.42  430 

9.99  985  8.42  446 

40 

30  18 .  37  484  9 .  99  988  8 .  37  497 

30 

30 

8.42   51019.99  985  8.42  525 

30 

40  8.37  573  9.99  988  8.37  585 

20 

40 

8.42  589!9.99  985  8.42  606 

20 

50  8.37  6629.99  988j8.37  674 

10 

50 

8.42  667 

9.99  985  8.42  683 

10 

22    0i8.37  75o'9.99  988'8.37  762 

0  38 

32    0 

8.42  746 

9.99  984  8.42  762 

0  28 

10 

8.37  838  9.99  988  8.37  850 

50 

10 

8.42  825 

9.99  984,8.42  840 

50 

20 

8.37  926  9.99  988  8.37  938 

40 

20 

8.42  903 

9.99  984  8.42  919 

40 

30 

8.38  014  9.99  987  8.38  026 

30 

30  18.42  982 

9.99  984 

8.42  997 

30 

40 

8.38  1019.99  987  8.38  114  20 

40  8.43  060 

9.99  984 

8.43  075 

20 

50 

8.38  189,9.99  987 

8.38  202 

10 

50 

8.43  138 

9.99  984 

8.43  154 

10 

23    0 

8.38  276^9.99  987 

8.38  289 

0  37 

33     0 

8.43  216 

9.99  984 

8.43  232 

0  27 

10  8.38  363  9.99  987  8.38  376  50 

10 

8.43  293 

9.99  984 

8.43  309 

50 

20  8.38  450  9.99  987  8.38  463  40 

20 

8.43  371 

9.99  984 

8.43  387 

40 

30  18.38  537  9.99  987S8.38  550:  30 

30 

8.43  448 

9.99  98418.43  464 

30 

40  i8.38  624  9.99  987 

8.38  636 

20 

40 

8.43  526 

9.99  984 

8.43  542 

20 

50i8.38  710|9.99  987 

8.38  723 

10 

50 

8.43  603 

9.99  984 

8.43  619 

10 

24    0  '8.38  796  9.99  987 

8.38  809 

0  36 

34    0 

8.43  680 

9.99  984 

8.43  696 

0  26 

10  8.38  882  9.99  987 

8.38  895 

50 

10 

8.43  757 

9.99  984  8.43  773 

50 

20  8.38  968  9.99  987 

8.38  981;  40 

20 

8.43  834 

9.99  98418.43  850 

40 

30  8 .  39  054  9 .  99  987 

8.39  067|  30 

30 

8.43  910 

9.99  984  8.43  927 

30 

40  |8.39  139  9.99  987 

8.39  153  20 

40 

8.43  987 

9.99  984 

£.44  003 

20 

50  8.39  225  9.99  987 

8.39  238 

10 

50 

8.44  063 

9.99  983 

8.44  080 

10 

25    0  '8.39  310  9.99  987 

8.39  323 

0  35 

35    0 

8.44  139 

9.99  983 

8.44  156 

0  25 

10  8.39  395J9.99  987 

8.39  408 

50 

10  18.44  216 

9.99  983 

8.44  232 

50 

20 

8 .  39  480  9 .  99  987  8 .  39  493i  40 

20  8.44  292 

9.99  983 

8.44  308 

40 

30 

8.39  565:9.99  987  8.39  587  30 

30  8.44  367 

9.99  983 

8.44  384 

30 

40 

8.39  649  9.99  987 

8.39  663i  20 

40  |8.44  443 

9.99  983 

8:44  460 

20 

50 

8.39  734 

9.99  986 

8.39  747 

10 

50  8.44  519 

9.99  983 

8.44  536 

10 

26    0 

8.39  818 

9.99  986 

8.39  832 

0  34 

36    0  18 .  44  594 

9.99  983 

8.44  611 

0  24 

10  8.39  902  9.99  986  8.39  916  50 

10  8 . 44  669 

9.99  983 

8.44  686 

50 

20  8.39  986  9.99  9868. 40  000  40 

20  8 .  44  745  9  .  99  983 

8.44  762 

40 

30  8 .  40  070  9 .  99  986  8 .  40  083'  30 

30  8 . 44  820 

9.99  983 

8.44  837 

30 

40 

8.40  153 

9.99  986 

8.40  167  20 

40 

8.44  895 

9.99  983 

8.44  912 

20 

50 

8.40  237 

9.99  986 

8.40  251 

10 

50 

8.44  969 

9.99  983 

8.44  987 

10 

27    0 

8.40  320 

9.99  986 

8.40  334 

0  33 

37    0 

8.45  044 

9.99  983 

8.45  061 

0  23 

10 

8.40  403 

9.99  986 

8.40  417 

50 

10 

8.45   119 

9.99  983 

8.45  136 

50 

20 

8.40  486 

9.99  98618.40  500  40 

20 

8.45   193  9.99  983 

8.45  210 

40 

30 

8.40  569  9.99  986:8.40  583  30 

30 

8.45  26719.99  983 

8.45  285 

30 

40 

8.40  651 

9.99  986 

8.40  665 

20 

40 

8.45  3419.99  982 

8.45  359 

20 

50 

8.40  734 

9.99  986 

8.40  748 

10 

50 

8.45  415 

9.99  982 

8.45  433 

10 

28    0 

8.40  816 

9.99  986 

8.40  830 

0  32 

38    0 

8.45  489 

9.99  982 

8.45  507 

0  22 

10 

8.40  898 

9.99  986 

8.40  913 

50 

10 

8.45  563 

9.99  982 

8.45  581 

50 

20 

8.40  980-9.99  986 

8 . 40  995  40 

20 

8.45  637  9.99  982|8.45  655 

40 

30 

8.41  062 

9.99  986 

8.41  077!  30 

30  8.45  7109.99  9828.45  728 

30 

40 

8.41    144 

9.99  986 

8.41   158  20 

40  8.45  784  9.99  982 

8.45  802 

20 

50 

8.41   225 

9.99  986 

8.41  240,  10 

50 

8.45  857,9.99  982 

8.45  875 

10 

29     0 

8.41  307 

9.99  985 

8.41   321     0  31 

39    0 

8.45  930  9.99  982 

8.45  948 

0  21 

10 

8.41  388  9.99  985|8.41  403  50 

10 

8.46  003  9.99  982  8.46  021 

50 

20 

8.41  469  9.99  9858.41  484  40 

20 

8 .  46  076  9.99  982  8 .  46  094 

40 

30 

8.41  550  9.99  985  8.41  565  30 

30 

8.46   149  9.99  982  8.46  167 

30 

40 

8.41  6319.99  985  8.41  646  20 

40 

8.46  222  9.99  982  8.46  240 

20 

50 

8.41  711j9.99  985 

8.41  726 

10 

50 

8.46  294  9.99  982|8.46  312 

1 

10 

30    0 

8.41  792 

9.99  985 

8.41  807 

0  30 

40    0 

8.46  366 

9.99  982 

8.40  385 

0  20 

/    // 

l 

LCos 

LSin 

LCot 

//    / 

I    n 

LCos 

LSin 

LCot 

//    / 

88°20'— Logarithms  of  Trigonometric  Functions— 88°40' 

— 10  should  be  written  after  every  logarithm  taken  from  this  page. 


II] 

1°40'- 

-Logarithms  of  Trigonometric  Functions 

—2° 

29 

/   // 

LSin 

LCos 

L  Tan !  "  ' 

'  "  j  L  Sin 

LCos 

LTan 

//  / 

40    0 

8.46  366 

9.99  982 

8.46  385|     0  20 

50    0  8.50  504 

9.99  978 

8.50  527 

0  10 

10  8.46  439  9.99  982  8.40  457  50 

10  8.50  570 

9.99  978  8.50  593 

50 

20 

8.46  51119.99  982J8.46  529  40 

20  8.50  636 

9.99  978  8.50  658 

40 

30 

8.46  5839.99  98118.46  602  30 

30  8.50  701 

9.99  978  8.50  724 

30 

40 

8.46  655|9.99  981  8.46  674  20 

40  8.50  767 

9 .  99  977  8 .  50  789 

20 

50 

8.46  72719.99  981  8.46  745 

10 

50,8.50  8329.99  977  8.50  855 

10 

41     0 

8.46  79919.99  98118.46  817 

0  19 

51     0  18.50  8979.99  977 

8.50  920 

0    9 

10  8.46  870  9.99  98l|8.46  889 

50 

10|8.50  963  9.99  977  8.50  985 

50 

20 

8.46  9429.99  981J8.46  960 

40 

20  8.51  028  9.99  977  8.51  050 

40 

30 

8.47^013  9.99  9818.47  032 

30 

30J8.51  092  9.99  977  8.51  015 

30 

40 

8.47  084  9.99  981 

8.47   103 

20 

40  8.51   157  9.99  977  8.51   180 

20 

50  8.47   155,9.99  981 

8.47  174 

10 

50 

8.51  222i9.99  977,8.51  245 

10 

42     0  8.47  226:9.99  981 

8.47  245 

0  18 

52     0 

8.51  287'9.99  9778.51  310 

0     8 

10 

8.47  29719.99  981 

8.47  316 

50 

10 

8.51  3519.99  977  8.51  374 

50 

20 

8.47  368,9.99  981 

8.47  387 

40 

20  8.51  416  9.99  977  8.51  439j  40 

30 

8.47  439  9.99  981J8.47  458 

30 

30  8.51  480  9.99  977  8.51  503    30 

40 

8.47  509 

9.99  981i8.47  528 

20 

40  8.51  544  9.99  977  8.51  568'  20 

50 

8.47  580 

9.99  981 

8.47  599 

10 

50  8.51  609,9.99  977.8.51  632 

10 

43    0 

8.47  650 

9.99  981 

8.47  669 

0  17 

53     0  8.51  673'9.99  977  8.51  696 

0     7 

10 

8.47  720 

9.99  980 

8.47  740 

50 

10  8.51  737|9.99  976  8.51  760 

50 

20 

8.47  790 

9.99  980 

8.47  810 

40 

20  8.51  8019.99  976  8.51   824 

40 

30 

8.47  860 

9.99  980  8.47  880| 

30 

30  8.51  864  9.99  976  8.51  888   30 

40 

8.47  930 

9.99  980 

8.47  950 

20 

40  8.51  928  9.99  976  8.51   952 

20 

50 

8.48  000 

9.99  980 

8.48  020 

10 

50  8.51  992  9.99  976 

8.52  015 

10 

44    0 

8.48  096 

9.99  980 

8.48  090 

0  16 

54    0  8.52  055  9.99  976 

8.52  079 

0    6 

10 

8.48  139J9.99  980 

8.48  159  50 

10 

8.52   119  9.99  976  8.52   143 

50 

20 

8.48  20819.99  980  8.48  228  40 

20 

8.52   1829.99  9768.52  206   40 

30 

8.48  278 

9.99  980  8.48  298  30 

30 

8.52  245  9.99  976  8.52  269 

30 

40 

8.48  347 

9.99  980 

8.48  367  20 

40 

8.52  3089.99  976 

8.52  332 

20 

50  8.48  416 

9.99  980 

8.48  436 

10 

50 

8.52  371  9.99  976 

8.52  396 

10 

45    0  8. 48  485 

9.99  980 

8.48  505 

0  15 

55     0 

8.52  434  9.99  976 

8.52  459 

0     5 

10  8.48  554 

9.99  980 

8.48  574 

50 

10  8.52  49719.99  976  8.52  522 

50 

20  8.48  622 

9.99  980 

8.48  643  40 

20  8.52  560|9.99  976 

8.52  584 

40 

30  8.48  691 

9.99  980 

8.48  711  30 

30  8.52  623  9.99  975 

8.52  647 

30 

40  8 . 48  760 

9.99  979 

8.48  780!  20 

40  8 .  52  685  9 .  99  975 

8.52  710 

20 

50 

8.48  828 

9.99  979 

8.48  849 

10 

50  8.52  748 

9.99  975 

8.52  772 

10 

46    0 

8.48  896 

9.99  979 

8.48  917 

0  14 

56    0'8.52  810 

9.99  975 

8.52  835 

0    4 

10 

8.48  965 

9.99  979 

8.48  985 

50 

10  8.52  872 

9.99  975 

8.52  897 

50 

20 

8.49  033 

9.99  979 

8.49  053 

40 

20  8.52  935  9.99  975 

8.52  960 

40 

30 

8.49   101 

9.99  979 

8.49   121 

30 

30  8.52  997  9.99  975 

8.53  022 

30 

40 

8.49  169 

9.99  979 

8.49   189 

20 

40 

8.53  059 

9.99  975 

8.53  084 

20 

50 

8.49  236 

9.99  979 

8.49  257 

10 

50 

8.53   121 

9.99  975 

8.53   146 

10 

47    0 

8.49  304 

9.99  979 

8.49  325 

0  13 

57     0 

8.53   183 

9.99  975 

8.53  208 

0    3 

10 

8.49  372 

9.99  979 

8.49  393 

50 

10 

8.53  245 

9.99  975 

8.53  270 

50 

20 

8.49  439 

9.99  979 

8.49  460 

40 

20 

8.53  306 

9.99  975 

8.53  332 

40 

30 

8.49  506 

9.99  979 

8.49  528 

30 

30 

8.53  368 

9.99  975  8.53  393 

30 

40  8.49  574 

9.99  979 

8.49  595 

20 

40 

8.53  429 

9.99  975 

8.53  455 

20 

50  8.49  641 

9.99  979 

8.49  662 

10 

50 

8.53  491 

9.99  974 

8.53  516 

10 

48    0  8.49  708 

9.99  979 

8.49  729 

0  12 

58     0 

8.53  552  9.99  974 

8.53  578 

0     2 

10 

8.49  775 

9.99  979 

8.49  796 

50 

10 

8.53  614  9.99  974  8.53  639 

50 

20 

8.49  842  9.99  978 

8.49  863 

40 

20  8.53  675  9.99  974  8.53  700 

40 

30 

8.49  908 

9.99  978 

8.49  930 

30 

30  8.53  736  9.99  974  8.53  762 

30 

40 

8.49  975 

9.99  978 

8.49  997 

20 

40  18.53  79719.99  974  8.53  823 

20 

50 

8.50  042 

9.99  978 

8.50  063 

10 

50 

8.53  858  9.99  974  8.53  884 

10 

49    0 

8.50  108 

9.99  978 

8.50  130 

0  11 

59     0 

8.53  919'9.99  974  8.53  945 

0     1 

10  8.50  174 

9.99  978 

8.50  196 

50 

10 

8.53  979  9.99  974|8.54  005 

50 

20  8.50  241 

9.99  978  8.50  263 

40 

20 

8.54  04019.99  9748.54  066 

40 

30  8.50  307 

9.99  97818.50  329  30 

30 

8.54  1019.99  974  8.54  127 

30 

40  8.50  373 

9.99  978  8.50  395  20 

40 

8.54   16119.99  974 

8.54  187 

20 

50  8.50  439 

9.99  978  8.50  461 

10 

50 

8.54  2229.99  974 

8.54  248 

10 

J   50    0 

8.50  504 

9.99  978 

8.50  527 

0  10 

60      0 

8.54  282  9.99  974 

8.54  308 

0    0 

LI 

LCos 

LSin 

L  Cot    " 

i   n 

L  Cos  |  L  Sin 

LCot 

//    / 

88°— Logarithms  of  Trigonometric  Functions— 88°20/ 

— 10  should  be  written  after  every  logarithm  taken  from  this  page. 


30 


2°— Logarithms  of  Trigonometric  Functions 


LSin 


LTan 


cd 


L  Cot  L  Cos 


PP 


o 

1 

2 
3 

4 

5 

6 

7 
8 
9 

lO 

11 
12 
13 
14 

15 

16 
17 
18 
19 

2© 

21 
22 
23 
24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 
48 
49 

50 

51 
52 
53 
54 

55 

56 

57 
58 
59 

60 


8.54  282 
8.54  642 

8.54  999 

8.55  354 

8.55  705 

8.56  054 
8.56  400 

8.56  743 

8.57  084 
8.57  421 

8.57  757 

8.58  089 
8.58  419 

8.58  747 

8.59  072 

8.59  395 

8.59  715 

8.60  033 
8.60  349 
8.60  662 

8.60  973 

8.61  282 
8.61  589 

8.61  894 

8.62  196 

8.62  497 

8.62  795 

8.63  091 
8.63  385 
8.63  678 

8.63  968 

8.64  256 
8.64  543 

8.64  827 

8.65  110 

8.65  391 
8.65  670 

8.65  947 

8.66  223 
8.66  497 

8.66  769 

8.67  039 
8.67  308 
8.67  575 

8.67  841 

8.68  104 
8.68  367 
8.68  627 

8.68  886 

8.69  144 

8.69  400 
8.69  654 

8.69  907 

8.70  159 
8.70  409 

8.70  658 

8.70  905 

8.71  151 
8.71  395 
8.71  638 

8.71  880 


360 
357 
355 
351 

349 

346 
343 
341 
337 

336 

332 
330 
328 
325 

323 

320 
318 
316 
313 

311 

309 
307 
305 
302 

301 

298 
296 
294 
293 

290 

288 
287 
284 
283 

281 

279 
277 
276 
274 

272 

270 
269 
267 
266 

263 

263 
260 
259 
258 

256 

254 
253 
252 
250 

249 

247 
246 
244 
243 

242 


8.54  308 

8.54  669 

8.55  027 
8.55  382 

8.55  734 

8.56  083 
8.56  429 

8.56  773 

8.57  114 
8.57  452 

8.57  788 

8.58  121 
8.58  451 

8.58  779 

8.59  105 

8.59  428 

8.59  749 

8.60  068 
8.60  384 

8.60  698 

8.61  009 
8.61  319 
8.61  626 

8.61  931 

8.62  234 

8.62  535 

8.62  834 

8.63  131 
8.63  426 

8.63  718 

8.64  009 
8.64  298 
8.64  585 

8.64  870 

8.65  154 

8.65  435 
8.65  715 

8.65  993 

8.66  269 
8.66  543 

8.66  816 

8.67  087 
8.67  356 
8.67  624 

8.67  890 

8.68  154 
8.68  417 
8.68  678 

8.68  938 

8.69  196 

8.69  453 
8.69  708 

8.69  962 

8.70  214 
8.70  465 

8.70  714 

8.70  962 

8.71  208 
8.71  453 
8.71  697 

8.71  940 


361 
358 
355 
352 

349 

346 
344 
341 
338 

336 

333 
330 
328 
326 

323 
321 
319 
316 
314 

311 

310 
307 
305 
303 

301 

299 
297 
295 
2*92 

291 

289 

287 
285 
284 

281 

280 
278 
276 
274 

273 

271 

269 
268 


264 

263 
261 
260 
258 

257 

255 
254 
252 
251 

249 

248 
246 
245 
244 

243 


1.45  692 
1.45  331 
1.44  973 
1.44  618 
1.44  266 

1.43  917 
1.43  571 
1.43  227 
1.42  886 
1.42  548 

1.42  212 
1.41  879 
1.41  549 
1.41  221 
1.40  895 

1.40  572 
1.40  251 
1.39  932 
1.39  616 
1.39  302 

1.38  991 
1.38  681 
1.38  374 
1.38  069 
1.37  766 

1.37  465 
1.37  166 
1.36  869 
1.36  574 
1.36  282 

1.35  991 
1.35  702 
1.35  415 
1.35  130 
1.34  846 

1.34  565 
1.34  285 
1.34  007 
1.33  731 
1.33  457 

1.33  184 
1.32  913 
1.32  644 
1.32  376 
1.32  110 

1.31  846 
1.31  583 
1.31  322 
1.31  062 
1.30  804 

1.30  547 
1.30  292 
1.30  038 
1.29  786 
1.29  535 

1.29  286 
1.29  038 
1.28  792 
1.28  547 
1.28  303 

1.28  060 


9.99  974 
9.99  973 
9.99  973 
9.99  972 
9.99  972 

9.99  971 
9.99  971 
9.99  970 
9.99  970 
9.99  969 

9.99  969 
9.99  968 
9.99  968 
9.99  967 
9.99  967 

9.99  967 
9.99  966 
9.99  966 
9.99  965 
9.99  964 

9.99  964 
9.99  963 
9.99  963 
9.99  962 
9.99  962 

9.99  961 
9.99  961 
9.99  960 
9.99  960 
9.99  959 

9.99  959 
9.99  958 
9.99  958 
9.99  957 
9.99  956 

9.99  956 
9.99  955 
9.99  955 
9.99  954 
9.99  954 

9.99  953 
9.99  952 
9.99  952 
9.99  951 
9.99  951 

9.99  950 
9.99  949 
9.99  949 
9.99  948 
9.99  948 

9.99  947 
9.99  946 
9.99  946 
9.99  945 
9.99  944 

9.99  944 
9.99  943 
9.99  942 
9.99  942 
9.99  941 

9.99  940 


<;c> 
89 

58 
57 
66 

55 

54 
53 
52 
51 

BO 

•19 
48 
47 
46 

15 

44 
43 
42 
41 

to 

39 
38 
37 
36 


34 
33 
32 
31 

SO 

29 

28 
27 
26 

25 

24 
23 
22 
21 

00 

19 
18 
17 
16 

15 

14 
13 
12 
11 

10 

9 


360      350     340 


II KJ 
1H6 
170 

201 


1 

36 

35 

:>. 

72 

70 

8 

108 

105 

4 

144 

140 

5 

180 

175 

a 

216 

210 

7 

252 

245 

S 

288 

280 

9 

324 

315 

330 

33 


272 
306 


320     310 


4 

132 

128 

R 

165 

160 

fl 

198 

192 

7 

231 

224 

R 

264 

256 

9 

297 

288 

31 
62 
93 
124 
155 
186 
217 
248 
279 


300      290     28S 

28.5 
57.0 
85.5 
114.0 
142.5 
171.0 
199.5 
228.0 
256.5 


280      275     270 


1 

30 

29 

2 

60 

5H 

8 

90 

87 

4 

120    J 

150   / 

116 

5 

145 

« 

180 

174 

7 

210 

203 

S 

240 

232 

9 

270 

261 

28.0 
56.0 
84.0 
112.0 
140.0 
168.0 
196.0 
224.0 
252.0 


27.5 
55.0 
82.5 
110.0 
137.5 
165.0 
192.5 
220.0 
247.5 


26.5 
53.0 
79.5 
106.0 
132.5 
159.0 
185.5 
212.0 
238.5 


26.0 
52.0 
78.0 
104.0 
130.0 
156.0 
182.0 
208.0 
234.0 


25.0 
50.0 
75.0 
100.0 
125.0 
150.0 
175.0 
200.0 
225.0 


24.5 
49.0 
73.5 
198.0 
122.5 
147.0 
171.5 
196.0 
220.5 


27.0 
54.0 
81.0 
108.0 
135.0 
162.0 
189.0 
216.0 
243.0 


265      260     255 


25.5 
51.0 
76.5 
102.0 
127.5 
153.0 
178.5 
204.0 
229.5 


250      245     240 


24.0 
48.0 
72.0 
96.0 
120.0 
144.0 
168.0 
192.0 
216.0 


L  Cos 


LCot 


cd 


LTan 


LSin 


PP 


87°— Logarithms  of  Trigonometric  Functions 

-10  should  be  written  after  every  logarithm  taken 

mns_ 


In  the  remaining  part  of  table  II 

frnin  t.h«  first-.    fl*v»nniT    n.nH  fnnrt.h  nnli 


3°— Logarithms  of  Trigonometric  Functions 


31 


LSin   d 


L  Tan  c  d  L  Cot 


LCos 


PP 


10 

11 

12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 
28 
29 

30 

31 
32 
33 

a4 

35 

36 
37 
38 
39 

40 

41 
42 
43 

44 

45 

46 

47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 

58 
59 

60 


8.71  880 

8.72  120 
8.72  359 
8.72  597 

8.72  834 

8.73  069 
8.73  303 
8.73  535 
8.73  767 

8.73  997 

8.74  226 
8.74  454 
8.74  680 

8.74  906 

8.75  130 

8.75  353 
8.75  575 

8.75  795 

8.76  015 
8.76  234 

8.76  451 
8.76  667 

8.76  883 

8.77  097 
8.77  310 

8.77  522 
8.77  733 

8.77  943 

8.78  152 
8.78  360 

8.78  568 
8.78  774 

8.78  979 

8.79  183 
8.79  386 

8.79  588 
8.79  789 

8.79  990 

8.80  189 
8.80  388 

8.80  585 
8.80  782 

8.80  978 

8.81  173 
8.81  367 

8.81  560 
8.81  752 

8.81  944 

8.82  134 
8.82  324 

8.82  513 
8.82  701 

8.82  888 

8.83  075 
8.83  261 

8.83  446 
8.83  630 
8.83  813 

8.83  996 

8.84  177 

8.84  358 


240 
239 
238 
237 

235 

234 
232 
232 
230 

229 

228 
226 
226 
224 

223 

222 
220 
220 
219 

217 

216 
216 
214 
213 

212 

211 
210 
209 
208 

208 

206 
205 
204 
203 

202 

201 
201 
199 
199 

197 

197 
196 
195 
194 

193 

192 
192 
190 
190 

189 

188 
187 
187 
186 

185 

184 
183 
183 
181 

181 


8.71  940 

8.72  181 
8.72  420 
8.72  659 

8.72  896 

8.73  132 
8.73  366 
8.73  600 

8.73  832 

8.74  063 

8.74  292 
8.74  521 
8.74  748 

8.74  974 

8.75  199 

8.75  423 
8,75  645 

8.75  867 

8.76  087 
8.76  306 

8.76  525 
8.76  742 

8.76  958 

8.77  173 
8.77  387 

8.77  600 

8.77  811 

8.78  022 
8.78  232 
8.78  441 

8.78  649 

8.78  855 

8.79  061 
8.79  266 
8.79  470 

8.79  673 

8.79  875 

8.80  076 
8.80  277 
8.80  476 

8.80  674 

8.80  872 

8.81  068 
8.81  264 
8.81  459 

8.81  653 

8.81  846 
•8.82  038 

8.82  230 
8.82  420 

8.82  610 
8.82  799 

8.82  987 

8.83  175 
8.83  361 

8.83  547 
8.83  732 

8.83  916 

8.84  100 
8.84  282 

8.84  464 


241 
239 
239 
237 

236 

234 
234 
232 
231 

229 

229 
227 
226 
225 

224 

,222 
222 
220 
219 

219 

217 
216 
215 
214 

213 

211 
211 
210 
209 

208 

206 
206 
205 
204 

203 

202 
201 
201 
199 

198 

198 
196 
196 
195 

194 

193 
192 
192 
190 
190 

189 
188 
188 
186 

186 

185 
184 
184 
182 

182 


1.28  060 
1.27  819 
1.27  580 
1.27  341 
1.27  104 

1.26  868 
1.26  634 
1.26  400 
1.26  168 
1.25  937 

1.25  708 
1.25  479 
1.25  252 
1.25  026 
1.24  801 

1.24  577 
1.24  355 
1.24  133 
1.23  913 
1.23  694 

1.23  475 
1.23  258 
1.23  042 
1.22  827 
1.22  613 

1.22  400 
1.22  189 
1.21  978 
1.21  768 
1.21  559 

1.21  351 
1.21  145 
1.20  939 
1.20  734 
1.20  530 

1.20  327 
1.20  125 
1.19  924 
1.19  723 
1.19  524 

1.19  326 
1.19  128 
1.18  932 
1.18  736 
1.18  541 


18  347 
18  154 
17  962 
17  770 
17  580 


1.17  390 
1.17  201 
1.17  013 
1.16  825 
1.16  639 

1.16  453 
1.16  268 
1.16  084 
1.15  900 
1.15  718 

1.15  536 


9.99  940 
9.99  940 
9.99  939 
9.99  938 
9.99  938 

9.99  937 
9.99  936 
9.99  936 
9.99  935 
9.99  934 

9.99  934 
9.99  933 
9.99  932 
9.99  932 
9.99  931 

9.99  930 
9.99  929 
9.99  929 
9.99  928 
9.99  927 

9.99  926 
9.99  926 
9.99  925 
9.99  924 
9.99  923 

9.99  923 
9.99  922 
9.99  921 
9.99  920 
9.99  920 

9.99  919 
9.99  918 
9.99  917 
9.99  917 
9.99  916 

9.99  915 
9.99  914 
9.99  913 
9.99  913 
9.99  912 

9.99  911 
9.99  910 
9.99  909 
9.99  909 
9.99  908 

9.99  907 
9.99  906 
9.99  905 
9.99  904 
9.99  904 

9.99  903 
9.99  902 
9.99  901 
9.99  900 
9.99  899 

9.99  898 
9.99  898 
9.99  897 
9.99  896 
9.99  895 

9.99  894 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 
48 
47 
46 

45 

44 
43 

42 
41 

40 

39 
38 
37 
36 

35 

34 

33 
32 
31 

30 

29 
28 

27 

26 

20 

24 

23 
22 
21 

20 

19 
18 
17 
16 

15 

14 
13 
12 

11 

10 

9 
8 

7 
0 


241  239  237  236  234 


1 

24.1    239    23.7    23.6    23.4 

:' 

48.2    47.8    47.4    47.2    46.8 

3 

72.3    71.7    71.1    70.8    70.2 

4 

96.4    95.6    94.8    94.4    93.6 

.> 

120.5  119.5  118.5  118.0  117.0 

« 

144.6  143.4  142.2  141.6  140.4 

7 

168.7  167.3  165.9  165.2  163.8 

H 

192.8  191.2  189.6  188.8  187.2 

9 

216.9  215.1  213.3  212.4  210.6 

232  231  229  227  226 

1 

23.2    23.1    22.9    22.7    22.6 

'..' 

46.4    46.2    45.8    45.4    45.2 

H 

69.6    69.3    68.7    68.1    67.8 

4 

92.8    92.4    91.6    90.8    90.4 

.-> 

116.0  115.5  114.5  113.5  113.0 

fi 

139.2  138.6  137.4  136.2  135.6 

7 

162.4  161.7  160.3  158.9  158.2 

8 

185.6  184.8  183.2  181.6  180.8 

9 

208.8  207.9  206.1  204.3  203.4 

224  222  220  219  217 


22.4 
44.8 
67.2 
89.6 
112.0 
134.4 
156.8 
179.2 
201.6 


22.2 
44.4 


2-2.0 
44.0 
66.6  66.0 
88.8  88.0 
111.0  110.0 
133.2  132.0 
155.4  154.0 
177.6  176.0 
199.8  198.0 


21.9  21.7 
43.8  43.4 
65.7  65.1 
87.6  86.8 
109.5  108.5 
131.4  130.2 
153.3  151.9 
175.2  173.6 
197.1  195.3 


216  214  213  211  209 

21.6  21.4  21.3  21.1  20.9 
43.2  42.8  42.6  ^2.2  41.8 
64.8  64.2  63.9  63.3  62.7 
86.4  85.6  85.2  84.4  83.6 
108.0  107.0  106.5  105.5  104.5 
129.6  128.4  127.8  126.6  125.4 
151.2  149.8  149.1  147.7  146.3 
172.8  171.2  170.4  168.8  167.2 
194.4  192.6  191.7  189.9  188.1 


208  206  203  201  199 

20.8  20.6  20.3  20.1  19.9 
41.6  41.2  40.6  40.2  39.8 
62.4  61.8  60.9  60.3  59.7 
83.2  82.4  81.2  80.4  79.6 
104.0  103.0  101.5  100.5  99.5 
124.8  1216  121.8  120.6  119.4 
145.6  144.2  142.1  140.7  139.3 
166.4  164.8  162.4  160.8  159.2 
187.2  185.4  182.7  180.9  179.1 


198  196  194  192  190 

19.8  19.6  19.4  19.2  19.0 
39.6  39.2  38.8  38.4  38.0 
59.4  58.8  58.2  57.6  57.0 
79.2  78.4  77.6  76.8  76.0 
99.0  98.0  97.0  96.0  95.0 
118.8  117.6  116.4  115.2  114.0 
138.6  137.2  135.8  134.4  133.0 
158.4  156.8  155.2  153.6  152.0 
178.2  176.4  174.6  172.8  171.0 


188  186  184  182  181 


18.8  18.6  18.4 
37.6  37.2  36.8 
56.4  55.8  55.2 
75.2  74.4  73.6 
94.0  93.0  92.0 
112.8  111.6  110.4 
131.6  130.2  128.8 


18.2  18.1 
36.4  36.2 
54.6  54.3 
72.8  72.4 
91.0  90.5 
109.2  108.6 
127.4  126.7 


150.4  148.8  147.2  145.6  144.8 
169.2  167.4  165.6  163.8  162.9 


LCos     d 


LCot 


cd    LTan 


LSin 


PP 


86°— Logarithms  of  Trigonometric  Functions 


32 

4° — Logarithms  of  Trigonometric  Functions                  III 

t 

LSin 

d 

LTan 

cd 

LCot 

L  Cos 

PP 

© 

8.84  358 

181 
179 
179 
178 

8.84  464 

182 
180 
180 
179 

1.15  536 

9.99  894 

oo 

1 

8.84  539 

8.84  646 

1.15  354 

9.99  893 

59 

182  181  180  179  178 

2 

8.84  718 

8.84  826 

1.15  174 

9.99  892 

58 

1 

18.2    18.1    18.0    17.9    17  8 

3 

8.84  897 

8.85  006 

1.14  994 

9.99  891 

57 

•> 

36.4    36.2    36.0    35.8    36.6 

4 

8.85  075 

8.85  185 

1.14  815 

9.99  891 

56 

3 

54.6    54.3    54.0    53.7    53.4 

177 

178 

4 

72.8    72.4    72.0    71.6    71.2 

5 

6 

8.85  252 
8.85  429 

177 
176 

8.85  363 
8.85  540 

-477 
177 

1.14  637 
1.14  460 

9.99  890 
9.99  889 

55 

54 

8 

7 

91.0    90.5    90.0    89.5    89.0 
109.2  108.6  108.0  107.4  106.8 
127.4  126.7  126.0  125.3  124.6 

7 

8.85  605 

175 
175 

8.85  717 

176 
176 

1.14  283 

9.99  888 

53 

8 

145.6  144.8  144.0  143.2  142.4 

8 

8.85  780 

8.85  893 

1.14  107 

9.99  887 

52 

9 

163.8  162.9  162.0  161.1  160.2 

9 

8.85  955 

8.86  069 

1.13  931 

9.99  886 

51 

io 

8.86  128 

173 

173 
173 
171 

8.86  243 

174 

174 
174 
172 
172 

1.13  757 

9.99  885 

50 

177  176  175  174  173 

11 

8.86  301 

8.86  417 

1.13  583 

9.99  884 

49 

1 

2 
3 

17.7    17.6    17.5    17.4    17.3 
35.4    85.2    35.0    34.8    34.6 
53.1    52  8    52  5    52  2    51  9 

12 

8.86  474 

8.86  591 

1.13  409 

9.99  883 

48 

13 

8.86  645 

171 

8.86  763 

1.13  237 

9.99  882 

47 

4 

70.8    70.4    70.0    69.6    69!  2 

14 

8.86  816 

8.86  935 

1.13  065 

9.99  881 

46 

■"> 

88,5    88.0    87.5    87.0    86.5 

171 

171 

6 

106.2  105.6  105.0  104.4  103.8 

15 

16 

8.86  987 

8.87  156 

169 
169 

8.87  106 
8.87  277 

171 
170 

1.12  894 
1.12  723 

9.99  880 
9.99  879 

45 

44 

7 

K 
9 

123.9  123.2  122.5  121.8  121.1 
141.6  140.8  140.0  139.2  138.4 
159.3  158.4  157.5  156.6  155.7 

17 

8.87  325 

169 
167 

8.87  447 

169 
169 

1.12  553 

9.99  879 

43 

18 

8.87  494 

8.87  616 

1.12  384 

9.99  878 

42 

19 

8.87  661 

8.87  785 

1.12  215 

9.99  877 

41 

172  171  170  169  168 

168 

168 

1 

17.2    17.1    17.0    16.9    16.8 

20 

8.87  829 

166 
166 
165 

8.87  953 

167 
167 
166 

1.12  047 

9.99  876 

40 

2 

34.4    34.2    34.0    33.8    33.6 

21 

8.87  995 

8.88  120 

1.11  880 

9.99  875 

39 

3 

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9.19  672 

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80°— Logarithms  of  Trigonometric  Functions 


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9.29  529 

9.30  391 

0.69  609 

9.99  137 

37 

3 

19.5  19.2  18.9 

24 

9.29  591 

9.30  457 

0.69  543 

9.99  135 

36 

4 

26.0  25.6  25.2 

63 

65 

3 

5 

32.5  32.0  31.5 

25 

9.29  654 

62 
63 
62 
62 

9.30  522 

65 
65 
65 
65 

0.69  478 

9.99  132 

2 
3 
3 
2 

35 

6 

39.0  38.4  37.8 

26 

9.29  716 

9.30  587 

0.69  413 

9.99  130 

34 

7 

45.5  44.8  44.1 

27 

9.29  779 

9.30  652 

0.69  348 

9.99  127 

33 

8 

52.0  51.2  50.4 

28 

9.29  841 

9.30  717 

0.69  283 

9.99  124 

32 

9 

58.5  57.6  56.7 

29 

9.29  903 

9.30  782 

0.69  218 

9.99  122 

31 

63 

64 

3 

30 

9.29  966 

62 
62 
61 
62 

9.30  846 

65 
64 
65 
64 

0.69  154 

9.99  119 

2 
3 
2 
3 

30 

31 

9.30  028 

9.30  911 

0.69  089 

9.99  117 

29 

32 

9.30  090 

9.30  975 

0.69  025 

9.99  114 

28 

33 

9.30  151 

9.31  040 

0.68  960 

9.99  112 

27 

34 

9.30  213 

9.31  104 

0.68  896 

9.99  109 

26 

62  61  60 

62 

64 

3 

35 

9.30  275 

61 
62 
61 
62 

9.31  168 

65 
64 
64 
64 

0.68  832 

9.99  106 

2 
3 
2 
3 

25 

1 

6.2  6.1  6.0 

36 

9.30  336 

9.31  233 

0.68  767 

9.99  104 

24 

2 

12.4  12.2  12.0 

37 

9.30  398 

9.31  297 

0.68  703 

9.99  101 

23 

3 

18.6  18.3  18.0 

38 

9.30  459 

9.31  361 

0.68  639 

9.99  099 

22 

4 

24.8  24.4  24.0 

39 

9.30  521 

9.31  425 

0.68  575 

9.99  096 

21 

5 

31.0  30.5  30.0 

61 

64 

3 

6 

37.2  36.6  36.0 

40 

9.30  582 

61 
61 
61 
61 

9.31  489 

63 
64 
63 

64 

0.68  511 

9.99  093 

2 
3 
2 
3 

20 

7 

43.4  42.7  42.0 

41 

9.30  643 

9.31  552 

0.68  448 

9.99  091 

19 

8 

49.6  48.8  48.0 

42 

9.30  704 

9.31  616 

0.68  384 

9.99  088 

18 

9 

55.8  54.9  54.0 

43 

9.30  765 

9.31  679 

0.68  321 

9.99  086 

17 

44 

9.30  826 

9.31  743 

0.68  257 

9.99  083 

16 

61 

63 

3 

45 

9.30  887 

60 
61 
60 

61 

9.31  806 

64 
63 
63 
63 

0.68  194 

9.99  080 

2 
3 
3 

2 

15 

46 

9.30  947 

9.31  870 

0.68  130 

9.99  078 

14 

47 

9.31  008 

9.31  933 

0.68  067 

9.99  075 

13 

48 

9.31  068 

9.31  996 

0.68  004 

9.99  072 

12 

59  3 

49 

9.31  129 

9.32  059 

0.67  941 

9.99  070 

11 

60 

63 

3 

1 

5.9  0.3 

50 

9.31  189 

61 
60 
60 
60 

9.32  122 

63 
63 
63 
62 

0.67  878 

9.99  067 

3 
2 
3 
3 

10 

2 

11.8  0.6 

51 

9.31  250 

9.32  185 

0.67  815 

9.99  064 

9 

3 

17.7  0.9 

52 

9.31  310 

9.32  248 

0.67  752 

9.99  062 

8 

4 

23.6  1.2 

53 

9.31  370 

9.32  311 

0.67  689 

9.99  059 

7 

5 

29.5  1.5 

54 

9.31  430 

9.32  373 

0.67  627 

9.99  056 

6 

6 

35.4  1.8 

60 

63 

2 

7 

41.3  2.1 

55 

9.31  490 

59 
60 
60 
59 

9.32  436 

62 
63 
62 
62 

0.67  564 

9.99  054 

3 
3 

2 
3 

5 

8 

47.2  2.4 

56 

9.31  549 

9.32  498 

0.67  502 

9.99  051 

4 

9 

53.1  2.7 

57 

9.31  609 

9.32  561 

0.67  439 

9.99  048 

3 

58 

9.31  669 

9.32  623 

0.67  377 

9.99  046 

2 

59 

9.31  728 

9.32  685 

0.67  315 

9.99  043 

1 

60 

62 

3 

60 

9.31  788 

9.32  747 

0.67  253 

9.99  040 

o 

LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

r 

PP 

78°— Logarithms  of  Trigonometric  Functions 


40 


12° — Logarithms  of  Trigonometric  Functions 


[H 


LSin 


d     L  Tan    c  d 


LCot 


LCos 


PP 


o 

1 

2 
3 
4 

5 

6 

7 
8 
9 

lO 

11 
12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 
28 
29 


31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 

42 
43 
44 

45 

46 

47 
48 
49 

50 

51 
52 
53 

54 

55 

56 
57 
58 
50 


9.31  788 
9.31  847 
9.31  907 

9.31  966 

9.32  025 

9.32  084 
9.32  143 
9.32  202 
9.32  261 
9.32  319 

9.32  378 
9.32  437 
9.32  495 
9.32  553 
9.32  612 

9.32  670 
9.32  728 
9.32  786 
9.32  844 
9.32  902 

9.32  960 

9.33  018 
9.33  075 
9.33  133 
9.33  190 

9.33  248 
9.33  305 
9.33  362 
9.33  420 
9.33  477 

9.33  534 
9.33  591 
9.33  647 
9.33  704 
9.33  761 

9.33  818 
9.33  874 
9.33  931 

9.33  987 

9.34  043 

9.34  100 
9.34  156 
9.34  212 
9.34  268 
9.34  324 

9.34  380 

9.34  436 

9.34  491 

9.34  547 

9.34  602 

9.34  658 
9.34  713 
9.34  769 
9.34  824 
9.34  879 

9.34  934 

9.34  989 

9.35  044 
9.35  009 
9.35  154 

9.35  209 


32  747 
32  810 
32  872 
32  933 

32  995 

33  057 
33  119 
33  180 
33  242 
33  303 

33  365 
33  426 
33  487 
33  548 
33  609 

33  670 
33  731 
33  792 
33  853 
33  913 

33  974 

34  034 
34  095 
34  155 
34  215 

34  276 
34  336 
34  396 
34  456 
34  516 

34  576 
34  635 
34  695 
34  755 
34  814 

34  874 
34  933 

34  992 

35  051 
35  111 

35  170 
.35  229 
.35  288 
.35  347 
.35  405 

35  464 
35  523 
35  581 
35  640 
35  698 

35  757 
35  815 
35  873 
35  931 

35  989 

36  047 
36  105 
36  163 

9.36  22  L 
9.36  279 

9.36  336 


63 
62 
61 
62 

62 
62 
61 
62 
61 

62 

61 
61 
61 

61 

61 

61 
61 
61 

60 

61 

60 
61 
60 

60 

61 

60 
60 
60 
60 

60 

59 

60 
60 
59 

60 

59 
59 
59 
60 

59 

59 
69 
59 

58 

59 

59 

58 
59 

58 

59 

58 
58 
58 
58 

58 

58 
58 
58 
58 

57 


0.67  253 
0.67  190 
0.67  128 
0.67  067 
0.67  005 

0.66  943 
0.66  881 
0.66  820 
0.66  7o8 
0.66  697 

0.66  635 
0.66  574 
0.66  513 
0.66  452 
0.66  391 

0.66  330 
0.66  269 
0.66  208 
0.66  147 
0.66  087 

0.66  026 
0.65  966 
0.65  905 
0.65  845 
0.65  785 

0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 

0.65  424 
0.65  365 
0.65  305 
0.65  245 
0.65  186 

0.65  126 
0.65  067 
0.65  008 
0.64  949 
0.64  889 

0.64  830 
0.64  771 
0.64  712 
0.64  653 
0.64  595 

0.64  536 

0.64  477 

0.64  419 

0.64  360 

0.64  302 

0.64  243 
0.64  185 
0.64  127 
0.64  069 
0.64  Oil 

0.63  953 
0.63  895 
0.63  837 
0.63  770 
0.63  721 

0.63  664 


9.99  040 
9.99  038 
9.99  035 
9.99  032 
9.99  030 

9.99  027 
9.99  024 
9.99  022 
9.99  019 
9.99  016 

9.99  013 
9.99  Oil 
9.99  008 
9.99  005 
9.99  002 

9.99  000 
9.98  997 
9.98  994 
9.98  991 
9.98  989 

9.98  986 
9.98  983 
9.98  980 
9.98  978 
9.98  975 

9.98  972 
9.98  969 
9.98  967 
9.98  964 
9.98  961 

9.98  958 
9.98  955 
9.98  953 
9.98  950 
9.98  947 

9.98  944 
9.98  941 
9.98  938 
9.98  936 
9.98  933 

9.98  930 
9.98  927 
9.98  924 
9.98  921 
9.98  919 

9.98  916 
9.98  913 
9.98  910 
9.98  907 
9.98  904 

9.98  901 
9.98  898 
9.98  896 
9.98  893 
9.98  890 

9.98  887 
9.98  884 
9.98  881 
9.98  878 
9.98  875 

9.98  872 


60 

59 
58 
57 
56 

55 

54 
53 
52 
51 

50 

49 

48 
47 
46 

45 

44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 

30 

29 
28 
27 
26 

25 

24 
23 
22 

21 

20 

19 
18 
17 
16 

15 

14 

13 
12 
11 

lO 

9 


63    62    61 

6.3    6.2    6.1 

12.6  12.4  12.2 
18.9  18.6  18.3 
25.2  24.8  24.4 
31.5  31.0  30.5 
37.8  37.2  36.6 
44.1  43.4  42.7 
50.4  49.6  48.8 

56.7  55.8  54. 


60    59 

6.0  5.9 
12.0  11.8 
18.0  17.7 
24.0  23.6 
30.0  29.5 
36.0  35.4 
42.0  41.3 
48.0  47.2 
54.0  53.1 


58     57 

5.8  5.7 
11.6  11.4 
17.4  17.1 
23.2  22.8 
29.0  28.5 
34.8  34.2 
40.6  39.9 
46.4  45.6 
52.2  51.3 


56    55      3 


5.6  5.5 
11.2  11.0 
16.8  16.5 
22.4  22.0 
28.0  27.5 
33.6  33.0 
39.2  38.5 
44.8  44.0 
50.4  49.5 


0.3 
0.6 
0.9 
1.2 
1.5 
l.S 
2.1 
LM 
2.7 


L  Cos      d      L  Cot     c  d    L  Tan      L  Sin 


PP 


77°— Logarithms  of  Trigonometric  Functions 


II] 


13°— Logarithms  of  Trigonometric  Functions 


41 


LSin 


d     L  Tan    c  d     L  Cot 


LCos 


PP 


o 

1 

e 

3 

4 

5 

6 

7 
8 
9 

lO 

11 
12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 

58 
59 

60 


9.35  209 
9.35  263 
9.35  318 
9.35  373 
9.35  427 

9.35  481 
9.35  536 
9.35  590 
9.35  644 
9.35  698 

9.35  752 
9.35  806 
9.35  860 
9.35  914 

9.35  968 

9.36  022 
9.36  075 
9.36  129 
9.36  182 
9.36  236 

9.36  289 
9.36  342 
9.36  395 
9.36  449 
9.36  502 

9.36  555 
9.36  608 
9.36  660 
9.36  713 
9.36  766 

9.36  819 
9.36  871 
9.36  924 

9.36  976 

9.37  028 

9.37  081 
9.37  133 
9.37  185 
9.37  237 
9.37  289 

9.37  341 
9.37  393 
9.37  445 
9.37  497 
9.37  549 

9.37  600 
9.37  652 
9.37  703 
9.37  755 
9.37  806 

9.37  858 
9.37  909 

9.37  960 

9.38  Oil 
9.38  062 

9.38  113 
9.38  164 
9.38  215 
9.38  266 
9.38  317 

9.38  368 


36  336 
36  394 
36  452 
36  509 
36  566 

36  624 
36  681 
36  738 
36  795 
36  852 

36  909 

36  966 

37  023 
37  080 
37  137 

37  193 
37  250 
37  306 
37  363 
37  419 

37  476 
37  532 
37  588 
37  644 
37  700 

37  756 
37  812 
37  868 
37  924 

37  980 

38  035 
38  091 
38  147 
38  202 
38  257 

38  313 
38  368 
38  423 
38  479 
38  534 

38  589 
38  644 
38  699 
38  754 
38  808 

38  863 
38  918 

38  972 

39  027 
39  082 

39  136 
39  190 
39  245 
39  299 
39  353 

39  407 
39  461 
.39  515 
.39  569 
.39  623 


9.39  677 


63  664 
63  606 
63  548 
63  491 
63  434 

63  376 
63  319 
63  262 
63  205 
63  148 

63  091 
63  034 
62  977 
62  920 
62  863 

62  807 
62  750 
62  694 
62  637 
62  581 

62  524 
62  468 
62  412 
62  356 
62  300 

62  244 
62  188 
62  132 
62  076 
62  020 

61  965 
61  909 
61  853 
61  798 
61  743 

61  687 
61  632 
61  577 
61  521 
61  466 

61  411 
61'  356 
61  301 
61  246 
61  192 

61  137 
61  082 
61  028 
60  973 
60  918 

60  864 
60  810 
60  755 
60  701 
60  647 

60  593 
60  539 
60  485 
60  431 
60  377 


9.98  872 
9.98  869 
9.98  867 
9.98  864 
9.98  861 

9.98  858 
9.98  855 
9.98  852 
9.98  849 
9.98  846 

9.98  843 
9.98  840 
9.98  837 
9.98  834 
9.98  831 


98  828 
98  825 
98  822 
98  819 
98  816 


0.60  323 


9.98  813 
9.98  810 
9.98  807 
9.98  804 
9.98  801 

9.98  798 
9.98  795 
9.98  792 
9.98  789 
9.98  786 

9.98  783 
9.98  780 
9.98  777 
9.98  774 
9.98  771 

9.98  768 
9.98  765 
9.98  762 
9.98  759 
9.98  756 

9.98  753 
9.98  750 
9.98  746 
9.98  743 
9.98  740 

9.98  737 
9.98  734 
9.98  731 
9.98  728 
9.98  725 

9.98  722 
9.98  719 
9.98  715 
9.98  712 
9.98  709, 

9.98  706 
9.98  703 
9.98  700 
9.98  697 
9.98  694 

9.98  690 


50 

58 

57 
56 

55 

54 
53 

52 
51 

SO 

49 

48 

47 
46 

15 

44 
43 
42 
41 

lO 

39 
38 

37 
36 

35 

34 
33 
32 
31 

80 

29 

28 
27 
20 

25 

24 
23 
22 
21 

20 

19 
18 

17 
16 

15 

14 
13 

12 
11 

10 

9 

8 
7 
6 

5 

4 
3 

2 
1 


58    57    56 

5.8  5.7  5.6 
11.6  11.4  11.2 
17.4  17.1  16.8 
23.2  22.8  22.4 
29.0  28.5  28.0 
34.8  34.2  33.6 
40.6  39.9  39.2 
46.6  45.6  44.8 
52.2  51.3  50.4 


55     54    53 

5.5  5.4  5.3 
11.0  10.8  10.6 
16.5  16.2  15.9 
22.0  21.6  21.2 
27.5  27.0  26.5 
33.0  32.4  31.8 
38.5  37.8  37.1 
44.0  43.2  42.4 
49.5  48.6  47.7 


52     SI 


5.2  5.1 
10.4  10.2 
15.6  15.3 
20.8  20.4 
26.0  25.5 
31.2  30.6 
36.4  35.7 
41.6  40.8 
46.8  45.9 


0.4  0.3 
0.8  0.6 
1.2    0.9 


1.2 


2.0  1.5 

2.4  1.8 

2.8  2.1 

3.2  2.4 

3.6  2.7 


LCos 


LCot 


cd 


LTan 


LSin 


PP 


76°— Logarithms  of  Trigonometric  Functions 


42 

14°— 

Logarithms  of  Trigonometric  Functions 

[II 

/ 

L  Sin 

d 

LTan 

cd 

LCot 

LCos 

d 

PP 

o 

9.38  368 

50 
51 
50 
51 

9.39  677 

54 
54 
53 
54 

0.60  323 

9.98  690 

3 
3 
3 
3 

64) 

1 

9.38  418 

9.39  731 

0.60  269 

9.98  687 

59 

2 

9.38  469 

9.39  785 

0.60  215 

9.98  684 

58 

3 

9.38  519 

9.39  838 

0.60  162 

9.98  681 

57 

4 

9.38  570 

9.39  892 

0.60  108 

9.98  678 

56 

50 

53 

3 

5 

9.38  620 

50 
51 
50 
50 

9.39  945 

54 
53 
54 
53 

0.60  055 

9.98  675 

4 
3 
3 
3 

55 

54  53 

6 

9.38  670 

9.39  999 

0.60  001 

9.98  671 

54 

7 

9.38  721 

9.40  052 

0.59  948 

9.98  668 

53 

1 

5.4  5.3 

8 

9.38  771 

9.40  106 

0.59  894 

9.98  665 

52 

2 

10.8  10.6 

9 

9.38  821 

9.40  159 

0.59  841 

9.98  662 

51 

3 

16.2  15.9 

50 

53 

3 

4 

21.6  21.2 

io 

9.38  871 

50 
50 
50 
50 

9.40  212 

54 
53 
53 
53 

0.59  788 

9.98  659 

3 

4 
3 
3 

50 

5 

27.0  26.5 

11 

9.38  921 

9.40  266 

0.59  734 

9 . 98  656 

49 

6 

32.4  31.8 

12 

9.38  971 

9.40  319 

0.59  681 

9.98  652 

48 

7 

37.8  37.1 

13 

9.39  021 

9.40  372 

0.59  628 

9.98  649 

47 

8 

43.2  42.4 

14 

9.39  071 

9.40  425 

0.59  575 

9.98  646 

46 

9 

48.6  47.7 

50 

53 

3 

15 

9.39  121 

49 
50 
50 
49 

9.40  478 

53 

53 
52 
53 

0.59  522 

9.98  643 

3 
4 
3 
3 

45 

16 

9.39  170 

9.40  531 

0.59  469 

9.98  640 

44 

17 

9.39  220 

9.40  584 

0.59  416 

9.98  636 

43 

18 

9.39  270 

9.40  636 

0.59  364 

9.98  633 

42 

19 

9.39  319 

9.40  689 

0.59  311 

9.98  630 

41 

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36 
36 
36 
36 

9.54  512 

40 
41 
40 
40 

0.45  488 

9.97  479 

4 
5 
4 
5 

40 

21 

9.52  027 

9.54  552 

0.45  448 

9.97  475 

39 

22 

9.52  063 

9.54  593 

0.45  407 

9.97  470 

38 

23 

9.52  099 

9.54  633 

0.45  367 

9.97  466 

37 

24 

9.52  135 

9.54  673 

0.45  327 

9.97  461 

36 

36 

41 

4 

25 

9.52  171 

36 
35 
36 
36 

9.54  714 

40 
40 
41 
40 

0.45  286 

9.97  457 

4 
5 
4 

35 

26 

9.52  207 

9.54  754 

0.45  246 

9.97  453 

34 

37  36  35 

27 

9.52  242 

9.54  794 

0.45  206 

9.97  448 

33 

28 

9.52  278 

9.54  835 

0.45  165 

9.97  444 

32 

1 

3.7  3.6  3.5 

29 

9.52  314 

9.54  875 

0.45  125 

9.97  439 

5 

31 

2 

7.4  7.2  7.0 

36 

40 

4 

3 

11.1  10.8  10.5 

30 

9.52  350 

35 
36 
35 
36 

9.54  915 

40 
40 
40 
40 

0.45  085 

9.97  435 

5 
4 
5 
4 

30 

4 

14.8  14.4  14.0 

31 

9.52  385 

9.54  955 

0.45  045 

9.97  430 

29 

5 

18.5  18.0  17.5 

32 

9.52  421 

9.54  995 

0.45  005 

9.97  426 

28 

6 

22.2  21.6  21.0 

33 

9.52  456 

9.55  035 

0.44  965 

9.97  421 

27 

7 

25.9  25.2  24.5 

34 

9.52  492 

9.55  075 

0.44  92g 

9.97  417 

26 

S 

29.6  28.8  28.0 

35 

40 

5 

9 

33.3  32.4  31.5 

35 

9.52  527 

36 
35 
36 
35 

9.55  115 

40 
40 
40 
40 

0.44  885 

9.97  412 

4 
5 
4 
5 

25 

36 

9.52  563 

9.55  155 

0.44  845 

9.97  408 

24 

37 

9.52  598 

9.55  195 

0.44  8O5 

9.97  403 

23 

38 

9.52  634 

9.55  235 

0.44  765 

9.97  399 

22 

39 

9.52  669 

9.55  275 

0.44  72g 

9.97  394 

21 

36 

40 

4 

40 

9.52  705 

35 
35 
36 
35 

9.55  315 

40 
40 
39 
40 

0.44  685 

9.97  390 

5 

4 
5 
4 

20 

41 

9.52  740 

9.55  355 

0.44  645 

9.97  385 

19 

42 

9.52  775 

9.55  395 

0.44  605 

9.97  381 

18 

43 

9.52  811 

9.55  434 

0.44  566 

9.97  376 

17 

44 

9.52  846 

9.55  474 

0.44  526 

9.97  372 

16 

34   5   4 

35 

40 

5 

45 

9.52  881 

35 
35 
35 
35 

9.55  514 

40 
39 
40 
40 

0.44  486 

9.97  367 

4 
5 
5 
4 

15 

1 

3.4  0.5  0.4 

46 

9.52  916 

9.55  554 

0.44  446 

9.97  363 

14 

2 

6.8  1.0  0.8 

47 

9.52  951 

9.55  593 

0.44  407 

9.97  358 

13 

3 

10.2  1.5  1.2 

48 

9.52  986 

9.55  633 

0.44  367 

9.97  353 

12 

4 

13.6  2.0  1.6 

49 

9.53  021 

9.55  673 

0.44  327 

9.97  349 

11 

5 

17.0  2.5  2.0 

35 

39 

5 

6 

20.4  3.0  2.4 

50 

9.53  056 

36 
34 
35 
35 

9.55  712 

40 
39 
40 
39 

0.44  288 

9.97  344 

4 
5 
4 
5 

IO 

7 

23.8  3.5  2.8 

51 

9.53  092 

9.55  752 

0.44  248 

9.97  340 

9 

8 

27.2  4.0  3.2 

52 

9.53  126 

9.55  791 

0.44  209 

9.97  335 

8 

9 

30.6  4.5  3.6 

53 

9.53  161 

9.55  831 

0.44  169 

9.97  331 

7 

54 

9.53  196 

9.55  870 

0.44  130 

9.97  326 

6 

35 

40 

4 

55 

9.53  231 

35 
35 
35 
34 

9.55  910 

39 
40 
39 
39 

0.44  090 

9.97  322 

5 
5 

4 
5 

5 

56 

9.53  266 

9.55  949 

0.44  051 

9.97  317 

4 

57 

9.53  301 

9.55  989 

0.44  Oil 

9.97  312 

3 

58 

9.53  336 

9.56  028 

0.43  972 

9.97  308 

2 

59 

9.53  370 

9.56  067 

0.43  933 

9.97  303 

1 

35 

40 

4 

60 

9.53  405 

9.56  107 

0.43  893 

9.97  299 

O 

LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

/ 

PP 

70°— -Logarithms  of  Trigonometric  Functions 


48 


20° — Logarithms  of  Trigonometric  Functions 


III 


/ 

LSin 

« 

LTan 

cd 

LCot 

LCos 

d 

PP 

o 

9.53  405 

35 
35 
34 
35 

9.56  107 

39 
39 
39 
40 

0.43  893 

9.97  299 

5 
5 

4 
5 

GO 

1 

9.53  440 

9.56  146 

0.43  854 

9.97  294 

59 

2 

9.53  475 

9.5.6  185 

0.43  815 

9.97  289 

58 

3 

9.53  509 

9.56  224 

0.43  776 

9.97  285 

57 

4 

9.53  544 

9.56  264 

0.43  736 

9.97  280 

56 

34 

39 

4 

5 

9.53  578 

35 
34 
35 
34 

9.56  303 

39 
39 
39 
39 

0.43  697 

9.97  276 

5 
5 
4 
5 

55 

6 

9.53  613 

9.56  342 

0.43  658 

9.97  271 

54 

7 

9.53  647 

9.56  381 

0.43  619 

9.97  266 

53 

8 

9.53  682 

9.56  420 

0.43  580 

9.97  262 

52 

9 

9.53  716 

9.56  459 

0.43  541 

9.97  257 

51 

40  39  38 

35 

39 

5 

io 

9.53  751 

34 
34 
35 
34 

9.56  498 

39 
39 
39 
39 

0.43  502 

9.97  252 

4 
5 
5 
4 

50 

1 

4.0  3.9  3.8 

11 

9.53  785 

9.56  537 

0.43  463 

9.97  248 

49 

2 

8.0  7.8  7.6 

12 

9.53  819 

9.56  576 

0.43  424 

9.97  243 

48 

3 

12.0  11.7  11.4 

13 

9.53  854 

9.56  615 

0.43  385 

9.97  238 

47 

4 

16.0  15.6  15.2 

14 

9.53  888 

9.56  654 

0.43  346 

9.97  234 

46 

5 

20.0  19.5  19.0 

34 

39 

5 

6 

24.0  23.4  22.8 

15 

9.53  922 

35 
34 
34 
34 

9.56  693 

39 
39 
39 
39 

0.43  307 

9.97  229 

5 
4 
5 
5 

45 

7 

28.0  27.3  26.6 

16 

9.53  957 

9.56  732 

0.43  268 

9.97  224 

44 

8 

32.0  31.2  30.4 

17 

9.53  991 

9.56  771 

0.43  229 

9.97  220 

43 

9 

36.0  35.1  34.2 

18 

9.54  025 

9.56  810 

0.43  190 

9.97  215 

42 

19 

9.54  059 

9.56  849 

0.43  151 

9.97  210 

41 

34 

38 

4 

20 

9.54  093 

34 
34 
34 
34 

9.56  887 

39 
39 
39 
38 

0.43  113 

9.97  206 

5 
5 
4 
5 

40 

21 

9.54  127 

9.56  926 

0.43  074 

9.97  201 

39 

22 

9.54  161 

9.56  965 

0.43  035 

9.97  196 

38 

23 

9.54  195 

9.57  004 

0.42  996 

9.97  192 

37 

24 

9.54  229 

9.57  042 

0.42  958 

9.97  187 

36 

34 

39 

5 

25 

9.54  263 

34 
34 
34 
34 

9.57  081 

39 
38 
39 
38 

0.42  919 

9.97  182 

4 
5 
5 
5 

35 

26 

9.54  297 

9.57  120 

0.42  880 

9.97  178 

34 

37  35  34 

27 

9.54  331 

9.57  158 

0.42  842 

9.97  173 

33 

28 

9.54  365 

9.57  197 

0.42  803 

9.97  168 

32 

1 

3.7  3.5  3.4 

29 

9.54  399 

9.57  235 

0.42  765 

9.97  163 

31 

2 

7.4  7.0  6.8 

34 

39 

4 

3 

11.1  10.5  10.2 

30 

9.54  433 

33 
34 
34 
33 

9.57  274 

38 
39 
38 
39 

0.42  726 

9.97  159 

5 
5 
4 
5 

30 

i 

14.8  14.0  13.6 

31 

9.54  466 

9.57  312 

0.42  688 

9.97  154 

29 

f) 

18.5  17.5  17.0 

32 

9.54  500 

9.57  351 

0.42  649 

9.97  149 

28 

G 

22.2  21.0  20.4 

33 

9.54  534 

9.57  389 

0.42  611 

9.97  145 

27 

7 

25.9  24.5  23.8 

34 

9.54  567 

9.57  428 

0.42  572 

9.97  140 

26 

8 

29.6  28.0  27.2 

34 

38 

5 

9 

33.3  31.5  30.6 

35 

9.54  601 

34 
33 
34 
33 

9.57  466 

38 
39 
38 

38 

0.42  534 

9.97  135 

6 

4 
5 
5 

25 

36 

9.54  635 

9.57  504 

0.42  496 

9.97  130 

24 

37 

9.54  668 

9.57  543 

0.42  457 

9.97  126 

23 

38 

9.54  702 

9.57  581 

0.42  419 

9.97  121 

22 

39 

9.54  735 

9.57  619 

0.42  381 

9.97  116 

21 

34 

39 

5 

40 

9.54  769 

33 
34 
33 
34 

9.57  658 

38 
38 
38 
38 

0.42  342 

9.97  111 

4 
5 
5 
5 

20 

41 

9.54  802 

9.57  696 

0.42  304 

9.97  107 

19 

42 

9.54  836 

9.57  734 

0.42  266 

9.97  102 

18 

43 

9.54  869 

9.57  772 

0.42  228 

9.97  097 

17 

44 

9.54  903 

9.57  810 

0.42  190 

9.97  092 

16 

33   5   4 

33 

39 

5 

45 

9.54  936 

33 
34 
33 
33 

9.57  849 

38 
38 
38 
38 

0.42  151 

9.97  087 

4 
5 
5 
5 

15 

1 

3.3  0.5  0.4 

46 

9.54  969 

9.57  887 

0.42  113 

9.97  083 

14 

2 

6.6  1.0  0.8 

47 

9.55  003 

9.57  925 

0.42  075 

9.97  078 

13 

8 

9.9  1.5  1.2 

48 

9.55  036 

9.57  963 

0.42  037 

9.97  073 

12 

-1 

13.2  2.0  1.6 

49 

9.55  069 

9.58  001 

0.41  999 

9.97  068 

11 

5 

16.5  2.5  2.0 

33 

38 

5 

0 

19.8  3.0  2.4 

50 

9.55  102 

34 
33 
33 
33 

9.58  039 

38 
38 
38 
38 

0.41  961 

9.97  063 

4 
5 
5 
5 

IO 

7 

23.1  3.5  2.8 

51 

9.55  136 

9.58  077 

0.41  923 

9.97  059 

9 

8 

26.4  4.0  3.2 

52 

9.55  L69 

9.58  115 

0.41  885 

9.97  054 

8 

9 

29.7  4.5  3.6 

53 

9.55  202 

9.58  153 

0.41  847 

9.97  049 

7 

54 

9.55  235 

9.58  191 

0.41  809 

9.97  044 

6 

33 

38 

5 

55 

9.55  268 

33 
33 
33 

33 

9.58  229 

38 
37 
38 
38 

0.41  771 

9.97  039 

4 
5 
5 
5 

5 

56 

9.66  301 

9.58  267 

0.41  733 

9.97  035 

4 

57 

9.66  334 

9.58  304 

0.41  696 

9.97  030 

3 

68 

9.66  367 

9.58  342 

0.41  658 

9.97  025 

2 

59 

9.55  400 

9.58  380 

0.41  620 

9.97  020 

1 

33 

38 

5 

GO 

9.55  433 

9.58  418 

0.41  582 

9.97  015 

O 

LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

/ 

PP 

69°— Logarithms  of  Trigonometric  Functions 


II] 


21°— Logarithms  of  Trigonometric  Functions 


49 


t 

LSin 

d 

LTan 

cd 

LCot 

L  Cos 

d 

PP 

o 

9.55  433 

33 
33 
33 
32 

9.58  418 

37 
38 
38 
38 

0.41  582 

9.97  015 

5 
5 

4 
5 

60 

1 

9.55  466 

9.58  455 

0.41  545 

9.97  010 

59 

2 

9.55  499 

9.58  493 

0.41  507 

9.97  005 

58 

3 

9.55  532 

9.58  531 

0.41  469 

9.97  001 

57 

4 

9.55  564 

9.58  569 

0.41  431 

9.96  996 

56 

33 

• 

37 

5 

5 

9.55  597 

33 
33 
32 
33 

9.58  606 

38 
37 
38 
38 

0.41  394 

9.96  991 

5 
5 
5 
5 

55 

6 

9.55  630 

9.58  644 

0.41  356 

9.96  986 

54 

7 

9.55  663 

9.58  681 

0.41  319 

9.96  981 

53 

8 

9.55  695 

9.58  719 

0.41  281 

9.96  976 

52 

9 

9.55  728 

9.58  757 

0,41  243 

9.96  871 

51 

38  37  36 

33 

37 

5 

io 

9.55  761 

32 
33 
32 
33 

9.58  794 

38 
37 
38 
37 

0.41  206 

9.96  966 

4 
5 
5 
5 

50 

1 

3.8  3.7  3.6 

11 

9.55  793 

9.58  832 

0.41  168 

9.96  962 

49 

2 

7.6  7.4  7.2 

12 

9.55  826 

9.58  869 

0.41  131 

9.96  957 

48 

3 

11.4  11.1  10.8 

13 

9.55  858 

9.58  907 

0.41  093 

9.96  952 

47 

4 

15.2  14.8  14.4 

14 

9.55  891 

9.58  944 

0.41  056 

9.96  947 

46 

5 

19.0  18.5  18.0 

32 

37 

5 

8 

22.8  22.2  21.6 

At 

9.55  923 

33 

32 
33 
32 

9.58  981 

38 
37 
38 
37 

0.41  019 

9.96  942 

5 
5 

5 
5 

45* 

7 

26.6  25.9  25.2 

9.55  956 

9.59  019 

0.40  981 

9.96  937 

44 

8 

30.4  29.6  28.8 

17 

9.55  988 

9.59  056 

0.40  944 

9.96  932 

43 

9 

34.2  33.3  32.4 

18 

9.56  021 

9.59  094 

0.40  906 

9.96  927 

42 

19 

9.56  053 

9.59  131 

0.40  869 

9.96  922 

41 

32 

37 

5 

20 

9.56  085 

33 
32 
32 
33 

9.59  168 

37 
38 
37 
37 

0.40  832 

9.96  917 

5 
5 
4 
5 

40 

21 

9.56  118 

9.59  205 

0.40  795 

9.96  912 

39 

22 

9.56  150 

9.59  243 

0.40  757 

9.96  907 

38 

23 

9.56  182 

9.59  280 

0.40  720 

9.96  903 

37 

24 

9.56  215 

9.59  317 

0.40  683 

9.96  898 

36 

32 

37 

5 

25 

9.56  247 

32 
32 
32 
32 

9.59  354 

37 
38 
37 
37 

0.40  646 

9.96  893 

5 
5 
5 
5 

35 

26 

9.56  279 

9.59  391 

0.40  609 

9.96  888 

34 

33  32  31 

27 

9.56  311 

9.59  429 

0.40  571 

9.96  883 

33 

28 

9.56  343 

9.59  466 

0.40  534 

9.96  878 

32 

1 

3.3  3.2  3.1 

29 

9.56  375 

9.59  503 

0.40  497 

9.96  873 

31 

2 

6.6  6.4  6.2 

33 

37 

5 

3 

9.9  9.6  9.3 

30 

9.56  408 

32 
32 
32 
32 

9.59  540 

37 
37 
37 
37 

0.40  460 

9.96  868 

5 
5 

5 
5 

30 

4 

13.2  12.8  12.4 

31 

9.56  440 

9.59  577 

0.40  423 

9.96  863 

29 

5 

16.5  16.0  15.5 

32 

9.56  472 

9.59  614 

0.40  386 

9.96  858 

28 

6 

19.8  19.2  18.6 

33 

9.56  504 

9.59  651 

0.40  349 

9.96  853 

27 

7 

23.1  22.4  21.7 

34 

9.56  536 

9.59  688 

0.40  312 

9.96  848 

26 

8 

26.4  25.6  24.8 

32 

37 

5 

9 

29.7  28.8  27.9 

35 

9.56  568 

31 
32 
32 
32 

9.59  725 

37 
37 
36 
37 

0.40  275 

9.96  843 

5 
5 
5 
5 

25 

36 

9.56  599 

9.59  762 

0.40  238 

9.96  838 

24 

37 

9.56  631 

9.59  799 

0.40  201 

9.96  833 

23 

38 

9.56  663 

9.59  835 

0.40  165 

9.96  828 

22 

39 

9.56  695 

9.59  872 

0.40  128 

9.96  823 

21 

32 

37 

5 

40 

9.56  727 

32 
31 
32 
32 

9.59  909 

37 
37 
36 
37 

0.40  091 

9.96  818 

b 

20 

41 

9.56  759 

9.59  946 

0.40  054 

9.96  813 

o 

19 

42 

9.56  790 

9.59  983 

0.40  017 

9.96  808 

5 
5 
5 

18 

43 

9.56  822 

9.60  019 

0.39  981 

9.96  803 

17 

44 

9.56  854 

9.60  056 

0.39  944 

9.96  798 

16 

6   5   4 

32 

37 

5 

45 

9.56  886 

31 
32 
31 
32 

9.60  093 

37 
36 
37 
37 

0.39  907 

9.96  793 

5 
5 
5 
6 

15 

1 

0.6  0.5  0.4 

46 

9.56  917 

9.60  130 

0.39  870 

9.96  788 

14 

2 

1.2  1.0  0.8 

47 

9.56  949 

9.60  166 

0.39  834 

9.96  783 

13 

3 

1.8  1.5  1.2 

48 

9.56  980 

9.60  203 

0.39  797 

9.96  778 

12 

4 

2.4  2.0  1.6 

49 

9.57  012 

9.60  240 

0.39  760 

9.96  772 

11 

5 

3.0  2.5  2.0 

32 

36 

5 

6 

3.6  3.0  2.4 

50 

9.57  044 

31 
32 
31 
31 

9.60  276 

37 
36 
37 
36 

0.39  724 

9.96  767 

5 
5 
5 
5 

IO 

7 

4.2  3.5  2.8 

51 

9.57  075 

9.60  313 

0.39  687 

9.96  762 

9 

8 

4.8  4.0  3.2 

52 

9.57  107 

9.60  349 

0.39  651 

9.96  757 

8 

9 

5.4  4.5  3.6 

53 

9.57  138 

9.60  386 

0.39  614 

9.96  752 

7 

54 

9.57  169 

9.60  422 

0.39  578 

9.96  747 

6 

32 

37 

5 

55 

9.57  201 

31 
32 
31 
31 

9.60  459 

36 
37 
36 
37 

0.39  541 

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5 
5 
5 
5 

5 

56 

9.57  232 

9.60  495 

0.39  505 

9.96  737 

4 

57 

9.57  264 

9.60  532 

0.39  468 

9.96  732 

3 

58 

9.57  295 

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2 

59 

9.57  326 

9.60  605 

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32 

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9.57  358 

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50 


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9.57  358 

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31 
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6 
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5 
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60 

1 

9.57  389 

9.60  677 

0.39  323 

9.96  711 

59 

2 

9.57  420 

9.60  714 

0.39  286 

9.96  706 

58 

3 

9.57  451 

9.60  750 

0.39  250 

9.96  701 

57 

4 

9.57  482 

9.60  786 

0.39  214 

9.96  696 

56 

32 

37 

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9.57  514 

31 
31 
31 
31 

9.60  823 

36 
36 
36 
36 

0.39  177 

9.96  691 

55 

6 

9.57  545 

9.60  859 

0.39  141 

9.96  686 

5 

54 

7 

9.57  576 

9.60  895 

0.39  105 

9.96  681 

5 
5 

6 

53 

8 

9.57  607 

9.60  931 

0.39  069 

9.96  676 

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9 

9.57  638 

9.60  967 

0.39  033 

9.96  670 

51 

37  36  35 

31 

37 

5 

io 

9.57  669 

31 
31 
31 
31 

9.61  004 

36 
36 
36 
36 

0.38  996 

9.96  665 

5 
5 
5 

50 

1 

3.7  3.6  3.5 

11 

9.57  700 

9.61  040 

0.38  960 

9.96  660 

49 

2 

7.4  7.2  7.0 

12 

9.57  731 

9.61  076 

0.38  924 

9.96  655 

48 

3 

11.1  10.8  10.5 

13 

9.57  762 

9.61  112 

0.38  888 

9.96  650 

47 

4 

14.8  14.4  14.0 

14 

9.57  793 

9.61  148 

0.38  852 

9.96  645 

5 

46 

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18.5  18.0  17.5 

31 

36 

5 

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22.2  21.6  21.0 

15 

9.57  824 

31 
30 
31 
31 

9.61  184 

36 
36 
36 
36 

0.38  816 

9.96  640 

6 
5 

45 

7 

25.9  25.2  24.5 

16 

9.57  855 

9.61  220 

0.38  780 

9.96  634 

44 

8 

29.6  28.8  28.0 

17 

9.57  885 

9.61  256 

0.38  744 

9.96  629 

43 

9 

33.3  32.4  31.5 

18 

9.57  916 

9.61  292 

0.38  708 

9.96  624 

5 

42 

19 

9.57  947 

9.61  328 

0.38  672 

9.96  619 

5 

41 

31 

36 

5 

2© 

9.57  978 

30 
31 
31 
31 

9.61  364 

36 
36 
36 
36 

0.38  636 

9.96  614 

6 
5 

40 

21 

9.58  008 

9.61  400 

0.38  600 

9.96  608 

39 

22 

9.58  039 

9.61  436 

0.38  564 

9.96  603 

38 

23 

9.58  070 

9.61  472 

0.38  528 

9.96  598 

5 

37 

24 

9.58  101 

9.61  508 

0.38  492 

9.96  593 

5 

36 

30 

«^ 

36 

5 

25 

9.58  131 

31 
30 
31 
30 

9.61  544 

35 
36 
36 
36 

0.38  456 

9.96  588 

6 
5 
5 
5 

35 

26 

9.58  162 

9.61  579 

0.38  421 

9.96  582 

34 

32  31  30 

27 

9.58  192 

9.61  615 

0.38  385 

9.96  577 

33 

28 

9.58  223 

9.61  651 

0.38  349 

9.96  572 

32 

1 

3.2  3.1  3.0 

29 

9.58  253 

9.61  687 

0.38  313 

9.96  567 

31 

2 

6.4  6.2  6.0 

31 

35 

5 

3 

9.6  9.3  9.0 

30 

9.58  284 

30 
31 
30 
31 

9.61  722 

36 
36 
36 
35 

0.38  278 

9.96  562 

6 
5 
5 
5 

30 

4 

12.8  12.4  12.0 

31 

9.58  314 

9.61  758 

0.38  242 

9.96  556 

29 

5 

16.0  15.5  15.0 

32 

9.58  345 

9.61  794 

0.38  206 

9.96  551 

28 

6 

19.2  18.6  18.0 

33 

9.58  375 

9.61  830 

0.38  170 

9.96  546 

27 

7 

22.4  21.7  21.0 

34 

9.58  406 

9.61  865 

0.38  135 

9.96  541 

26 

8 

25,6  24.8  24.0 

30 

36 

6 

9 

28.8  27.9  27.0 

35 

9.58  436 

31 
30 
30 
30 

9.61  901 

35 
36 
36 
35 

0.38  099 

9.96  535 

5 
5 
5 
6 

25 

36 

9.58  467 

9.61  936 

0.38  064 

9.96  530 

24 

37 

9.58  497 

9.61  972 

0.38  028 

9.96  525 

23 

38 

9.58  527 

9.62  008 

0.37  992 

9.96  520 

22 

39 

9,58  557 

9.62  043 

0.37  957 

9.96  514 

21 

31 

36 

5 

40 

9.58  588 

30 
30 
30 
31 

9.62  079 

35 
36 
35 
36 

0.37  921 

9.96  509 

5 
6 
5 
5 

20 

41 

9.58  618 

9.62  114 

0.37  886 

9.96  504 

19 

42 

9.58  648 

9.62  150 

0.37  850 

9.96  498 

18 

43 

9.58  678 

9.62  185 

0.37  815 

9.96  493 

17 

44 

9.58  709 

9.62  221 

0.37  779 

9.96  488 

16 

29   6   5 

30 

35 

5 

45 

9.58  739 

30 
30 
30 
30 

9.62  256 

36 
35 
35 
36 

0.37  744 

9.96  483 

6 
5 
5 
6 

15 

1 

2.9  0.6  0.5 

46 

9.58  769 

9.62  292 

0.37  708 

9.96  477 

14 

2 

5.8  1.2  1.0 

47 

9.58  799 

9.62  327 

0.37  673 

9.96  472 

13 

3 

8.7  1.8  1.5 

48 

9.58  829 

9.62  362 

0.37  638 

9.96  467 

12 

4 

11.6  2.4  2.0 

49 

9.58  859 

9.62  398 

0.37  602 

9.96  461 

11 

s 

14.5  3.0  2.5 

30 

35 

5 

6 

17.4  3.6  3.0 

50 

9.58  889 

30 
30 
30 
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9.62  433 

35 
36 
35 
35 

0.37  567 

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6 
5 
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IO 

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20.3  4.2  3.5 

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9.58  919 

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8 

23.2  4.8  4.0 

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9.58  949 

9.62  504 

0.37  496 

9.96  445 

8 

9 

26.1  5.4  4.5 

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9.58  979 

9.62  539 

0.37  461 

9.96  440 

7 

54 

9.59  009 

9.62  574 

0.37  426 

9.96  435 

6 

30 

35 

6 

55 

9.59  039 

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29 
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9.62  609 

36 
35 
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5 
5 
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56 

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57 

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58 

9.59  128 

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2 

59 

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0.37  250 

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1 

30 

35 

5 

60 

9.59  188 

9.62  785 

0.37  215 

9.96  403 

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PP 

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51 

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L  Sin 

d 

LTan 

cd 

LCot 

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PP 

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9.59  188 

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35 
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6 
5 
5 
6 

60 

1 

9.59  218 

9.62  820 

0.37  180 

9.96  397 

59 

o 

9.59  247 

9.62  855 

0.37  145 

9.96  392 

58 

3 

9.59  277 

9.62  890 

0.37  110 

9.96  387 

57 

4 

9.59  307 

9.62  926 

0.37  074 

9.96  381 

56 

29 

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9.59  336 

30 
30 
29 
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9.62  961 

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35 
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6 
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5 
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55 

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9.59  366 

9.62  996 

0.37  004 

9.96  370 

54 

7 

9.59  396 

9.63  031 

0.36  969 

9.96  365 

53 

8 

9.59  425 

9.63  066 

0.36  934 

9.96  360 

52 

9 

9.59  455 

9.63  101 

0.36  899 

9.96  354 

51 

36  35  34 

29 

34 

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io 

9.59  484 

30 
29 
30 
29 

9.63  135 

35 
35 
35 
35 

0.36  865 

9.96  349 

6 
5 
5 
6 

50 

1 

3.6  3.5  3.4 

11 

9.59  514 

9.63  170 

0.36  830 

9.96  343 

49 

2 

7.2  7.0  6.8 

12 

9.59  543 

9.63  205 

0.36  795 

9.96  338 

48 

3 

10.8  10.5  10.2 

13 

9.59  573 

9.63  240 

0.36  760 

9.96  333 

47 

4 

14.4  14.0  13.6 

14 

9.59  602 

9.63  275 

0.36  725 

9.96  327 

46 

5 

18.0  17.5  17.0 

30 

35 

5 

6 

21.6  21.0  20.4 

15 

9.59  632 

29 
29 
30 
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9.63  310 

35 
34 
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35 

0.36  690 

9.96  322 

6 
5 
6 
5 

45 

7 

25.2  24.5  23.8 

16 

9.59  661 

9.63  345 

0.36  655 

9.96  316 

44 

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28.8  28.0  27.2 

17 

9.59  690 

9.63  379 

0.36  621 

9.96  311 

43 

9 

32.4  31.5  30.6 

18 

9.59  720 

9.63  414 

0.36  586 

9.96  305 

42 

19 

9.59  749 

9.63  449 

0.36  551 

9.96  300 

41 

29 

35 

6 

20 

9.59  778 

30 
29 
29 
29 

9.63  484 

35 

34 
35 
35 

0.36  516 

9.96  294 

5 
5 
6 
5 

40 

21 

9.59  808 

9.63  519 

0.36  481 

9.96  289 

39 

22 

9.59  837 

9.63  553 

0.36  447 

9.96  284 

38 

23 

9.59  866 

9.63  588 

0.36  412 

9.96  278 

37 

24 

9.59  895 

9.63  623 

0.36  377 

9.96  273 

36 

29 

34 

6 

25 

9.59  924 

30 
29 
29 
29 

9.63  657 

35 
34 
35 
35 

0.36  343 

9.96  267 

5 
6 
5 
6 

35 

26 

9.59  954 

9.63  692 

0.36  308 

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34 

30  2?  28 

27 

9.59  983 

9.63  726 

0.36  274 

9.96  256 

33 

28 

9.60  012 

9.63  761 

0.36  239 

9.96  251 

32 

1 

3.0  2.9  2.8 

29 

9.60  041 

9.63  796 

0.36  204 

9.96  245 

31 

2 

6.0  5.8  5.6 

29 

34 

5 

3 

9.0  8.7  8.4 

30 

9.60  070 

29 
29 
29 
29 

9.63  830 

35 
34 
35 
34 

0.36  170 

9.96  240 

6 
5 
6 
5 

30 

4 

12.0  11.6  11.2 

31 

9.60  099 

9.63  865 

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29 

5 

15.0  14.5  14.0 

32 

9.60  128 

9.63  899 

0.36  101 

9.96  229 

28 

6 

18.0  17.4  16.8 

33 

9.60  157 

9.63  934 

0.36  066 

9.96  223 

27 

7  21.0  20.3  19.6 

34 

9.60  186 

9.63  968 

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26 

8  24.0  23.2  22.4 

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35 

6 

9  27.0  26.1  25.2 

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9.60  215 

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29 
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9.64  003 

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5 
6 
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25 

36 

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37 

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23 

38 

9.60  302 

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39 

9.60  331 

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21 

28 

35 

5 

40 

9.60  359 

29 
29 
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28 

9.64  175 

34 
34 
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34 

0.35  825- 

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20 

41 

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42 

9.60  417 

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18 

43 

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6 
6 

17 

44 

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6  5 

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34 

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29 
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29 

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34 
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34 

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6 
5 
6 
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15 

1 

0.6  0.5 

46 

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14 

2 

1.2  1.0 

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13 

3 

1.8  1.5 

48 

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12 

4 

2.4  2.0 

49 

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11 

5 

3.0  2.5 

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34 

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6 

3.6  3.0 

50 

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29 

28 
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34 
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6 
5 
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7 

4.2  3.5 

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8 

4.8  4.0 

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9.64  586 

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9 

5.4  4.5 

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9.60  732 

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54 

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28 

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55 

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28 
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28 

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34 
34 
34 
34 

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6 
5 
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56 

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4 

57 

9.60  846 

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3 

58 

9.60  875 

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59 

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cd 

LTan 

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d 

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PP 

66°— Logarithms  of  Trigonometric  Functions 


52 


24°— Logarithms  of  Trigonometric  Functions 


/ 

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d 

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cd 

LCot 

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9.60  931 

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28 
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34 
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GO 

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2 

9.60  988 

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58 

3 

9.61  016 

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57 

4 

9.61  045 

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34 
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55 

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40 

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9.61  522 

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23 

9.61  578 

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9.61  606 

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29  28  27 

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6 
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64°— Logarithms  of  Trigonometric  Functions 


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63°— Logarithms  of  Trigonometric  Functions 


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57 


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23.2  22.4  21.6 

17 

9.72  763 

9.80  056 

0.19  944 

9.92  707 

43 

9 

26.1  25.2  24.3 

18 

9.72  783 

9.80  084 

0.19  916 

9.92  699 

42 

19 

9.72  803 

9.80  112 

0.19  888 

9.92  691 

41 

^ 

20 

28 

.8 

20 

9.72  823 

20 
20 
20 
19 

9.80  140 

28 
27 
28 
28 

0.19  860 

9.92  683 

8 
8 
8 

8 

40 

21 

9.72  843 

9.80  168 

0.19  832 

9.92  675 

39 

22 

9.72  863 

9.80  195 

0.19  805 

9.92  667 

38 

23 

9.72  883 

9.80  223 

0.19  777 

9.92  659 

37 

24 

9.72  902 

9.80  251 

0.19  749 

9.92  651 

36 

20 

28 

8 

25 

9.72  922 

20 
20 
20 
20 

9.80  279 

28 
28 
28 
28 

0.19  721 

9.92  643 

8 
8 
8 

8 

35 

26 

9.72  942 

9.80  307 

0.19  693 

9.92  635 

34 

21  20  19 

27 

9.72  962 

9.80  335 

0.19  665 

9.92  627 

33 

28 

9.72  982 

9.80  363 

0.19  637 

9.92  619 

32 

1 

2.1  2.0  1.9 

29 

9.73  002 

9.80  391 

0.19  609 

9.92  611 

31 

2 

4.2  4.0  3.8 

20 

28 

8 

3 

6.3  6.0  5.7 

SO 

9.73  022 

19 
20 
20 
20 

9.80  419 

28 
27 

28 
28 

0.19  581 

9.92  603 

8 
8 
8 
8 

30 

4 

8.4  8.0  7.6 

31 

9.73  041 

9.80  447 

0.19  553 

9.92  595 

29 

5 

10.5  10.0  9.5 

32 

9.73  061 

9.80  474 

0.19  526 

9.92  587 

28 

6 

12.6  12.0  11.4 

33 

9.73  081 

9.80  502 

0.19  498 

9.92  579 

27 

7 

14.7  14.0  13.3 

34 

9.73  101 

9.80  530 

0.19  470 

9.92  571 

26 

8 

16.8  16.0  15.2 

20 

28 

8 

9 

18.9  18.0  17.1 

35 

9.73  121 

19 
20 
20 
20 

9.80  558 

28 

28 
28 
27 

0.19  442 

9.92  563 

8 
9 
8 
8 

25 

36 

9.73  140 

9.80  586 

0.19  414 

9.92  555 

24 

37 

9.73  160 

9.80  614 

0.19  386 

9.92  546 

23 

38 

9.73  180 

9.80  642 

0.19  358 

9.92  538 

22 

39 

9.73  200 

9.80  669 

0.19  331 

9.92  530 

21 

19 

28 

8 

40 

9.73  219 

20 
20 
19 
20 

9.80  697 

28 
28 
28 
27 

0.19  303 

9.92  522 

8 
8 
8 
8 

20 

41 

9.73  239 

9.80  725 

0.19  275 

9.92  514 

19 

42 

9.73  259 

9.80  753 

0.19  247 

9.92  506 

18 

43 

9.73  278 

9.80  781 

0.19  219 

9.92  498 

17 

44 

9.73  298 

9.80  808 

0.19  192 

9.92  490 

16 

9   8   7 

20 

28 

8 

45 

9.73  318 

19 
20 
20 
19 

9.80  836 

28 
28 
27 
28 

0.19  164 

9.92  482 

9 

8 
8 
8 

15 

1 

0.9  0.8  0.7 

46 

9.73  337 

9.80  864 

0.19  136 

9.92  473 

14 

2 

1.8  1.6  1.4 

47 

9.73  357 

9.80  892 

0.19  108 

9.92  465 

13 

3 

2.7  2.4  2.1 

48 

9.73  377 

9.80  919 

0.19  081 

9.92  457 

12 

4 

3.6  3.2  2.8 

49 

9.73  396 

9.80  947 

0.19  053 

9.92  449 

11 

5 

4.5  4.0  3.5 

20 

28 

8 

6 

5.4  4.8  4.2 

50 

9.73  416 

19 
20 
19 
20 

9.80  975 

28 
27 
28 
28 

0.19  025 

9.92  441 

8 
8 
9 

IO 

7 

6.3  5.6  4.9 

51 

9.73  435 

9.81  003 

0.18  997 

9.92  433 

9 

8 

7.2  6.4  5.6 

52 

9.73  455 

9.81  030 

0.18  970 

9.92  425 

8 

9 

8.1  7.2  6.3 

53 

9.73  474 

9.81  058 

0.18  942 

9.92  416 

7 

54 

9.73  494 

9.81  086 

0.18  914 

9.92  408 

8 

6 

19 

27 

8 

55 

9.73  513 

20 
19 
20 

19 

9.81  113 

28 
28 
27 
28 

0.18  887 

9.92  400 

8 
8 
8 
9 

5 

56 

9.73  533 

9.81  141 

0.18  859 

9.92  392 

4 

57 

9.73  552 

9.81  169 

0.18  831 

9.92  384 

3 

58 

9.73  572 

9.81  196 

0.18  804 

9.92  376 

2 

59 

9.73  591 

9.81  224 

0.18  776 

9.92  367 

1 

20 

28 

9 

60 

9.73  611 

9.81  252 

0.18  748 

9.92  359 

O 

LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

/ 

PP 

57°— Logarithms  of  Trigonometric  Functions 


II] 


33° — Logarithms  of  Trigonometric  Functions 


61 


LSin 


LTan 


cd 


LCot 


LCos 


PP 


o 

1 

2 
3 
4 

5 

6 

7 
8 
9 

lO 

11 
12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 

24 

25 

26 
27 

28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 
48 
49 

50 

51 
52 
53 

54 

55 

56 
57 
58 
59 

60 


9.73  611 
9.73  630 
9.73  650 
9.73  669 
9.73  689 

9.73  708 
9.73  727 
9.73  747 
9.73  766 
9.73  785 

9.73  805 
9.73  824 
9.73  843 
9.73  863 
9.73  882 

9.73  901 
9.73  921 
9.73  940 
9.73  959 
9.73  978 

9.73  997 

9.74  017 
9.74  036 
9.74  055 
9.74  074 

9.74  093 
9.74  113 
9.74  132 
9.74  151 
9.74  170 

9.74  189 
9.74  208 
9.74  227 
9.74  246 
9.74  265 

9.74  284 
9.74  303 
9.74  322 
9.74  341 
9.74  360 

9.74  379 
9.74  398 
9.74  417 
9.74  436 
9.74  455 

9.74  474 
9.74  493 
9.74  512 
9.74  531 
9.74  549 

9.74  568 
9.74  587 
9.74  606 
9.74  625 
9.74  644 

9.74  662 
9.74  681 
9.74  700 
9.74  719 
9.74  737 

9.74  756 


.81  252 
.81  279 
.81  307 
.81  335 
.81  362 

.81  390 
.81  418 
.81  445 
.81  473 
.81  500 

81  528 
.81  556 
81  583 
81  611 
81  638 

81  666 
81  693 
81  721 
,81  748 
81  776 

81  803 
81  831 
81  858 
81  886 
81  913 

81  941 
81  968 

81  996 

82  023 
82  051 

82  078 
82  106 
82  133 
82  161 
82  188 

82  215 
82  243 
82  270 
82  298 
82  325 

82  352 
82  380 
82  407 
82  435 
82  462 

82  489 
82  517 
82  544 
82  571 
82  599 

82  626 
82  653 
82  681 
82  708 
82  735 

82  762 
82  790 

82  817 
82  844 
82  871 


0.18  748 
0.18  721 
0.18  693 
0.18  665 
0.18  638 


0.18  610 
0.18  582 
0.18  555 
0.18  527 
0.18  500 


0.18  472 
0.18  444 
0.18  417 
0.18  389 
0.18  362 

0.18  334 
0.18  307 
0.18  279 
0.18  252 
0.18  224 

0.18  197 
0.18  169 
Q. 18  142 
0.18  114 
0.18  087 

0.18  059 
0.18  032 
0.18  004 
0.17  977 
0.17  949 

0.17  922 
0.17  894 
0.17  867 
0.17  839 
0.17  812 

0.17  785 
0.17  757 
0.17  730 
0.17  702 
0.17  675 


17  648 
17  620 
17  593 
17  565 
17  538 


0.17  511 
0.17  483 
0.17  456 
0.17  429 
0.17  401 


17  374 
17  347 
17  319 
17  292 
17  265 


0.17  238 
0.17  210 
0.17  183 
0.17  156 
0.17  129 


9.92  359 
9.92  351 
9.92  343 
9.92  335 
9.92  326 

9.92  318 
9.92  310 
9.92  302 
9.92  293 
9.92  285 

9.92  277 
9.92  269 
9.92  260 
9.92  252 
9.92  244 

9.92  235 
9.92  227 
9.92  219 
9.92  211 
9.92  202 


9.92 
9.92 
9.92 
9.92 
9.92 


194 
186 
177 
169 
161 


9.82  899 


0.17  101 


9.92  152 
9.92  144 
9.92  136 
9.92  127 
9.92  119 

9.92  111 
9.92  102 
9.92  094 
9.92  086 
9.92  077 

9.92  069 
9.92  060 
9.92  052 
9.92  044 
9.92  035 

9.92  027 
9.92  018 
9.92  010 
9.92  002 
9.91  993 

9.91  985 
9.91  976 
9.91  968 
9.91  959 
9.91  951 

9.91  942 
9.91  934 
9.91  925 
9.91  917 
9.91  908 

9.91  900 
9.91  891 
9.91  883 
9.91  874 
9.91  866 

9.91  857 


60 

59 
58 

57 
50 

55 

54 
53 
52 
51 

BO 

49 
4S 

47 
46 

15 

44 
43 
42 
41 

lO 

39 
38 
37 
36 

SB 

34 
33 
32 
31 

SO 

29 
28 
27 
26 

25 

24 
23 
22 
21 

99 

19 
18 
17 
16 

15 

14 
13 
12 
11 

lO 

9 
8 
7 
6 

5 

4 
3 
2 
1 


28     27 

2.8  2.7 

5.6  5.4 

8.4  8.1 

11.2  10.8 

14.0  13.5 

16.8  16.2 

19.6  18.9 

22.4  21.6 

25.2  24.3 


20    19    18 


2.0 
4.0 
6.0 
8.0 
10.0 


l.S 


1.9 

3.8 

5.7 

7.6 

9.5 
12.0  11.4 
14.0  13.3  12.6 
16.0  15.2  14.4 
18.0  17.1  16.2 


5.4 
7.2 
9.0 
10.8 


0.9 
1.8 
2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 


0.8 
1.6 
2.4 
3.2 
4.0 
4.8 
5.6 
6.4 
7.2 


LCos 


LCot     cd 


LTan 


LSin 


PP 


56° — Logarithms  of  Trigonometric  Functions 


62 


34°— Logarithms  of  Trigonometric  Functions 


LSin 


LTan    cd 


LCot 


LCos 


7 
8 
9 

lO 

11 
12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 
27 

28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 

58 
59 


9.74  756 
9.74  775 
9.74  794 
9.74  812 
9.74  831 

9.74  850 
9.74  868 
9.74  887 
9.74  906 
9.74  924 

9.74  943 
9.74  961 
9.74  980 

9.74  999 

9.75  017 

9.75  036 
9.75  054 
9.75  073 
9.75  091 
9.75  110 

9.75  128 
9.75  147 
9.75  165 
9.75  184 
9.75  202 

9.75  221 
9.75  239 
9.75  258 
9.75  276 
9.75  294 

9.75  313 
9.75  331 
9.75  350 
9.75  368 
9.75  386 

9.75  405 
9.75  423 
9.75  441 
9.75  459 
9.75  478 

9.75  496 
9.75  514 
9.75  533 
9.75  551 
9.75  569 

9.75  587 
9.75  605 
9.75  624 
9.75  642 
9.75  660 

9.75  678 
9.75  696 
9.75  714 
9.75  733 
9.75  751 

9.75  769 
9.75  787 
9.75  805 
9.75  823 
9.75  841 

9.75  859 


82  899 
82  926 
82  953 

82  980 

83  008 

83  035 
83  062 
83  089 
83  117 
83  144 

83  171 
83  198 
83  225 
83  252 
83  280 

83  307 
83  334 
83  361 
83  388 
83  415 

83  442 
83  470 
83  497 
83  524 
83  551 

83  578 
83  605 
83  632 
83  659 
83  686 

83  713 

83  740 
83  768 
83  795 
83  822 

83  849 
83  876 
83  903 
83  930 
83  957 

83  984 

84  Oil 
84  038 
84  065 
84  092 

84  119 
84  146 
84  173 
84  200 

84  227 

84  254 
84  280 
84  307 
84  334 
84  361 

84  388 
84  415 
84  442 
84  469 
84  496 


9.84  523 


0.17  101 
0.17  074 
0.17  047 
0.17  020 
0.16  992 

0.16  965 
0.16  938 
0.16  911 
0.16  883 
0.16  856 

0.16  829 
0.16  802 
0.16  775 
0.16  748 
0.16  720 

0.16  693 
0.16  666 
0.16  639 
0.16  612 
0.16  585 

0.16  558 
0.16  530 
0.16  503 
0.16*476 
0.16  449 

0.16  422 
0.16  395 
0.16  368 
0.16  341 
0.16  314 

0.16  287 
0.16  260 
0.16  232 
0.16  205 
0.16  178 

0.16  151 
0.16  124 
0.16  097 
0.16  070 
0.16  043 

0.16  016 
0.15  989 
0.15  962 
0.15  935 
0.15  908 

0.15  881 
0.15  854 
0.15  827 
0.15  800 
0.15  773 

0.15  746 
0.15  720 
0.15  693 
0.15  666 
0.15  639 

0.15  612 

0.15  585 
0.15  558 
0.15  531 
0.15  504 

0.15  477 


9.91  857 
9.91  849 
9.91  840 
9.91  832 
9.91  823 

9.91  815 
9.91  806 
9.91  798 
9.91  789 
9.91  781 

9.91  772 
9.91  763 
9.91  755 
9.91  746 
9.91  738 

9.91  729 
9.91  720 
9.91  712 
9.91  703 
9.91  695 

9.91  686 
9.91  677 
9.91  669 
9.91  660 
9.91  651 

9.91  643 
9.91  634 
9.91  625 
9.91  617 
9.91  608 

9.91  599 
9.91  591 
9.91  582 
9.91  573 
9.91  565 

9.91  556 
9.91  547 
9.91  538 
9.91  530 
9.91  521 

9.91  512 
9.91  504 
9.91  495 
9.91  486 
9.91  477 

9.91  469 
9.91  460 
9.91  451 
9.91  442 
9.91  433 

9.91  425 
9.91  416 
9.91  407 
9.91  398 
9.91  389 

9.91  381 
9.91  372 
9.91  363 
9.91  354 
9.91  345 

9.91  336 


59 

58 

57 
56 

55 

54 
53 

52 
51 

ISO 

49 
48 

47 
46 

15 

44 

43 
42 
41 

40 

39 
38 

37 
36 

85 

34 
33 
32 
31 

80 

29 

28 
27 
26 

25 

24 
23 
22 
21 

SO 

1!) 
IS 

17 
16 

18 

14 
13 
12 
LI 

10 

9 

s 

7 
6 

8 

4 
3 

2 

1 


L  Cos 


LCot 


cd 


LTan 


L  Sin 


55°— Logarithms  of  Trigonometric  Functions 


II] 

35°— 

Logarithms  of  Trigonometric  Functions 

63 

1 

LSin 

d 

LTan 

cd 

LCot 

LCos 

d 

PP 

© 

9.75  859 

18 
18 
18 
18 

9.84  523 

27 
26 
27 
27 

0.15  477 

9.91   336 

8 
9 
9 
9 

60 

1 

9.75  877 

9.84  550 

0.15  450 

9.91  328 

59 

2 

9.75  895 

9.84  576 

0.15  424 

9.91  319 

58 

3 

9.75  913 

9.84  603 

0.15  397 

9.91  310 

57 

4 

9.75  931 

9.84  630 

0.15  370 

9.91  301 

56 

18 

27 

9 

5 

9.75  949 

18 
18 
18 
18 

9.84  657 

27 
27 
27 
26 

0.15  343 

9.91  292 

9 
9 

8 
9 

55 

6 

9.75  967 

9.84  684 

0.15  316 

9.91  283 

54 

7 

9.75  985 

9.84  711 

0.15  289 

9.91  274 

53 

8 

9.76  003 

9.84  738 

0.15  262 

9.91  266 

52 

9 

9.76 -021 

9.84  764 

0.15  236 

9.91  257 

51 

27    26 

18 

27 

9 

io 

9.76  039 

18 
18 
18 
18 

9.84  791 

27 

27 
27 
27 

0.15  209 

9.91  248 

9 
9 
9 
9 

50 

1 

2.7    2.6 

11 

9.76  057 

9.84  818 

0.15   182 

9.91  239 

49 

2 

5.4    5.2 

12 

9.76  075 

9.84  845 

0.15   155 

9.91  230 

48 

3 

'8.1    7.8 

13 

9.76  093 

9.84  872 

0.15   128 

9.91   221 

47 

4 

10.8  10.4 

14 

9.76   111 

9.84  899 

0.15   101 

9.91  212 

46 

5 

13.5  13.0 

18 

26 

9 

6 

16.2  15.6 

15 

9.76   129 

17 
18 
18 
18 

9.84  925 

27 
27 

27 
27 

0.15  075 

9.91  203 

9 
9 
9 
9 

45 

7 

18.9  18.2 

16 

9.76  146 

9.84  952 

0.15  048 

9.91   194 

44 

8 

21.6  20.8 

17 

9.76   164 

9.84  979 

0.15  021 

9.91   185 

43 

9 

24.3  23.4 

18 

9.76   182 

9.85  006 

0.14  994 

9.91   176 

42 

19 

9.76  200 

9.85  033 

0.14  967 

9.91   167 

41 

18 

26 

9 

20 

9.76  218 

18 
17 

18 
18 

9.85  059 

27 
27 
27 
26 

0.14  941 

9.91   158 

9 
8 
9 
9 

40 

21 

9.76  236 

9.85  086 

0.14  914 

9.91    149 

39 

22 

9.76  253 

9.85   113 

0.14  887 

9.91   141 

38 

23 

9.76  271 

9.85   140 

0.14  860 

9.91    132 

37 

24 

9.76  289 

9.85   166 

0.14  834 

9.91   123 

36 

18 

27 

9 

25 

9.76  307 

17 

18 
18 
18 

9.85   193 

27 
27 
26 

27 

0.14  807 

9.91   114 

9 
9 
9 
9 

35 

26 

9.76  324 

9.85  220 

0.14  780 

9.91   105 

34 

18     17 

27 

9.76  342 

9.85  247 

0.14  753 

9.91  096 

33 

28 

9.76  360 

9.85  273 

0.14  727 

9.91  087 

32 

1 

1.8    1.7 

29 

9.76  378 

9.85  300 

0.14  700 

9.91  078 

31 

2 

3.6    3.4 

17 

27 

9 

3 

5.4    5.1 

30 

9.76  395 

18 
18 
17 
18 

9.85  327 

27 
26 
27 

27 

0.14  673 

9.91  069 

9 
9 
9 

9 

30 

4 

7.2    6.8 

31 

9.76  413 

9.85  354 

0.14  646 

9.91  060 

29 

5 

9.0    8.5 

32 

9.76  431 

9.85  380 

0.14  620 

9.91  051 

28 

6 

10.8  10.2 

33 

9.76  448 

9.85  407 

0.14  593 

9.91  042 

27 

7 

12.6  11.9 

34 

9.76  466 

9.85  434 

0.14  566 

9.91  033 

26 

8 

14.4  13.6 

18 

26 

10 

9 

16.2  15.3 

35 

9.76  484 

17 
18 
18 
17 

9.85  460 

27 
27 

26 

27 

0.14  540 

9.91  023 

9 
9 
9 
9 

25 

36 

9.76  501 

9.85  487 

0.14  513 

9.91  014 

24 

37 

9.76  519 

9.85  514 

0.14  486 

9.91  005 

23 

38 

9.76  537 

9.85  540 

0.14  460 

9.90  996 

22 

39 

9.76  554 

9.85  567 

0.14  433 

9.90  987 

21 

18 

27 

9 

40 

9.76  572 

18 
17 
18 
17 

9.85  594 

26 
27 
27 
26 

0.14  406 

9.90  978 

9 
9 
9 
9 

20 

41 

9.76  590 

9.85  620 

0.14  380 

9.90  969- 

19 

42 

9.76  607 

9.85  647 

0.14  353 

9.90  960 

.18 

43 

9.76  625 

9.85  674 

0.14  326 

9.90  951 

17 

44 

9.76  642 

9.85  700 

0.14  300 

9.90  942 

16 

10      9       8 

18 

27 

9 

45 

9.76  660 

17 
18 
17 
18 

9.85  727 

27 
26 

27 
27 

0.14  273 

9.90  933 

9 
9 
9 

15 

1 

1.0    0.9    0.8 

46 

9.76  677 

9.85  754 

0.14  246 

9.90  924 

14 

2 

2.0     1.8     1.6 

47 

9.76  695 

9.85  780 

0.14  220 

9.90  915 

13 

3 

3.0    2.7    2.4 

48 

9.76  712 

9.85  807 

0.14   193 

9.90  906 

12 

4 

4.0    3.6    3.2 

49 

9.76  730 

9.85  834 

0.14   166 

9.90  896 

10 

11 

5 

5.0    4.5    4.0 

17 

26 

9 

6 

6.0    5.4    4.8 

50 

9.76  747 

18 
17 
18 
17 

9.85  860 

27 
26 
27 
27 

0.14   140 

9.90  887 

9 
9 
9 
9 

IO 

7 

7.0    6.3    5.6 

51 

9.76  765 

9.85  887 

0.14   113 

9.90  878 

9 

8 

8.0    7.2    6.4 

52 

9.76  782 

9.85  913 

0.14  087 

9.90  869 

8 

9 

9.0    8.1    7.2 

53 

9.76  800 

9.85  940 

0.14  060 

9.90  860 

7 

54 

9.76  817 

9.85  967 

0.14  033 

9.90  851 

6 

18 

26 

9 

55 

9.76  835 

17 
18 
17 
17 

9.85  993 

27 
26 

27 

27 

0.14  007 

9.90  842 

10 
9 
9 

5 

56 

9.76  852 

9.86  020 

0.13  980 

9.90  832 

4 

57 

9.76  870 

9.86  046 

0.13  954 

9.90  823 

3 

58 

9.76  887 

9.86  073 

0.13  927 

9.90  814 

2 

59 

9.76  904 

9.86  100 

0.13  900 

9.90  805 

9 

1 

18 

26 

9 

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9.76  922 

9.86  126 

0.13  874 

9.90  796 

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LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

/ 

PP 

54 

t°— ] 

[rfOgarithn 

is  ol 

:  Trigonoi 

metric  Fu 

inct] 

ions 

64 

36° — Logarithms  of  Trigonometric  Functions 

[II 

/ 

LSin 

d 

LTan 

cd 

LCot 

LCos 

d 

PP 

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9.76  922 

17 
18 
17 
17 

9.86   126 

27 
26 
27 
26 

0.13  874 

9.90  796 

9 

10 

9 

9 

60 

1 

9,.  76  039 

9.86   153 

0.13  847 

9.90  787 

59 

2 

9.76  957 

9.86   179 

0.13  821 

9,90  777 

58 

3 

9.76  974 

9.86  206 

0.13  794 

9.90  768 

57 

4 

9.76  991 

9.86  232 

0.13  768 

9.90  759 

56 

■ 

18 

27 

9 

5 

9.77  009 

17 
17 
18 
17 

9.86  259 

26 

27 
26 
27 

0.13  741 

9.90  750 

9 

10 
9 
9 

55 

6 

9.77  026 

9.86  285 

0.13  715 

9.90  741 

54 

7 

9.77  043 

9.86  312 

0.13  688 

9.90  731 

53 

8 

9.77  061 

9.86  338 

0.13  662 

9.90  722 

52 

9 

9.77  078 

9.86  365 

0.13  635 

9.90  713 

51 

27    26 

17 

27 

9 

* 

io 

9.77  095 

17 

18 
17 
17 

9.86  392 

26 

27 
26 

27 

0.13  608 

9.90  704 

10 
9 
9 
9 

50 

1 

2.7    2.6 

11 

9.77   112 

9.86  418 

0.13  582 

9.90  694 

49 

2 

5.4    5.2 

12 

9.77   130 

9.86  445 

0.13  555 

9.90  685 

48 

3 

8.1    7.8 

13 

9.77   147 

9.86  471 

0.13  529 

9.90  676 

47 

4 

10.8  10.4 

14 

9.77   164 

9.86  498 

0.13  502 

9.90  667 

46 

5 

13.5  13.0 

17 

26 

10 

6 

16.2  15.6      » 

15 

9.77  181 

18 
17 
17 
17 

9.86  524 

27 
26 
26 
27 

0.13  476 

9.90  657 

9 

9 

9 

10 

45 

7 

18.9  18.2 

16 

9.77  199 

9.86  551 

0.13  449 

9.90  648 

44 

8 

21.6  20.8 

17 

9.77  216 

9.86  577 

0.13  423 

9.90  639 

43 

9 

24.3  23.4 

18 

9.77  233 

9.86  603 

0.13  397 

9.90  630 

42 

19 

9.77  250 

9.86  630 

0.13  370 

9.90  620 

41 

18 

26 

9 

-. 

20 

9.77  268 

17 
17 
17 
17 

9.86  656 

27 
26 
27 
26 

0.13  344 

9.90  611 

9 

10 

9 

9 

40 

21 

9.77  285 

9.86  683 

0.13  317 

9.90  602 

39 

22 

9.77  302 

9.86  709 

0.13  291 

9.90  592 

38 

23 

9.77  319 

9.86  736 

0.13  264 

9.90  583 

37 

24 

9.77  336 

9.86  762 

0.13  238 

9.90  574 

36 

17 

27 

9 

25 

9.77  353 

17 

17 
18 
17 

9.86  789 

26 
27 
26 
26 

0.13  211 

9.90  565 

10 
9 
9 

10 

35 

18     17     16  ' 

26 

9.77  370 

9.86  815 

0.13   185 

9.90  555 

34 

27 

9.77  387 

9.86  842 

0.13   158 

9.90  546 

33 

28 

9.77  405 

9.86  868 

0.13   132 

9.90  537 

32 

1 

1.8    1.7    1.6 

29 

9.77  422 

9.86  894 

0.13   106 

9.90  527 

31 

2 

3.6    3.4    3.2 

17 

27 

9 

3 

5.4    5.1    4.8 

30 

9.77  439 

17 
17 
17 
17 

9.86  921 

26 
27 
26 
27 

0.13  079 

9.90  518 

9 
10 

9 
10 

30 

4 

7.2    6.8    6.4 

31 

9.77  456 

9.86  947 

0.13  053 

9.90  509 

29 

5 

9.0    8.5    8.0 

32 

9.77  473 

9.86  974 

0.13  026 

9.90  499 

28 

6 

10.8  10.2    9.6 

33 

9.77  490 

9.87  000 

0.13  000 

9.90  490 

27 

: 

12.6  11.9  11.2 

34 

9.77  507 

9.87  027 

0.12  973 

9.90  480 

26 

8 

14.4  13.6  12.8 

17 

26 

9 

9 

16.2  15.3  14.4  k 

35 

9.77  524 

17 
17 
17 

17 

9.87  053 

26 
27 
26 
26 

0.12  947 

9.90  471 

9 

10 

9 

9 

25 

36 

9.77  541 

9.87  079 

0.12  921 

9.90  462 

24 

i 

37 

9.77  558 

9.87   106 

0.12  894 

9 . 90  452 

23 

38 

9.77  575 

9.87   132 

0.12  868 

9.90  443 

22 

1 

39 

9.77  592 

9.87   158 

0.12  842 

9.90  434 

21 

1 

17 

27 

10 

40 

9.77  609 

17 
17 
17 
17 

9.87   185 

26 

27 
26 
26 

0.12  815 

9.90  424 

9 
10 

9 
10 

20 

i 

41 

9.77  626 

9.87  211 

0.12  789 

9.90  415 

19 

42 

9.77  643 

9.87  238 

0.12  762 

9.90  405 

18 

43 

9.77  660 

9.87  264 

0.12  736 

9.90  396 

17 

1 

44 

9.77  677 

9.87  290 

0.12  710 

9.90  386 

16 

10      9 

17 

27 

9 

45 

9.77  694 

17 
17 
16 
17 

9.87  317 

26 
26 

27 
26 

0.12  683 

9.90  377 

9 
10 

9 
10 

15 

1 

1.0    0.9 

46 

9.77  711 

9.87  343 

0.12  657 

9.90  368 

14 

2 

2.0     1.8 

47 

9.77  728 

9.87   369 

0.12  631 

9.90  358 

13 

3 

3.0    2.7 

48 

9.77  744 

9.87   396 

0.12  604 

9.90  349 

12 

4 

4.0    3.6 

49 

9.77  761 

9.87  422 

0.12  578 

9.90  339 

11 

5 

5.0    4.5 

17 

26 

9 

6 

6.0    5.4 

50 

9.77  778 

17 
17 
17 
17 

9.87  448 

■  27 
26 
26 

27 

0.12  552 

9.90  330 

10 
9 

10 
9 

IO 

7 

7.0    6.3 

51 

9.77  795 

9.87  475 

0.12  525 

9.90  320 

9 

8 

8.0    7.2 

52 

9.77  812 

9.87  501 

0.12  499 

9.90  311 

8 

9 

9.0    8.1 

53 

9.77  829 

9.87   527 

0.12  473 

9.90  301 

7 

54 

9.77  846 

9.87  554 

0.12  446 

9.90  292 

6 

16 

26 

10 

55 

9.77  862 

17 
17 
17 
17 

9.87  580 

26 
27 
26 
26 

0.12  420 

9.90  282 

9 
10 

9 
10 

5 

56 

9.77  879 

9.87  606 

0.12  394 

9.90  273 

4 

57 

9.77  896 

9.87  633 

0.12  367 

9.90  263 

3 

58 

9.77  913 

9.87  659 

0.12  341 

9.90  254 

2 

59 

9.77  930 

9.87  685 

0.12  315 

9.90  244 

1 

16 

26 

9 

60 

9.77  946 

9.87  711 

0.12  289 

9.90  235 

O 

LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

/ 

PP 

5; 

3°— 

Logarithrj 

as  o 

f  Trigono 

metric  Fi 

met 

ions 

II]  37°— Logarithms  of  Trigonometric  Functions 


65  . 


LSin 

d 

LTan 

cd 

L  Cot 

L  Cos  j 

d 

PP 

o 

9.77  946 

17 
17 
17 
16 

9.87  711 

27 
26 
26 

27 

0.12  289 

9.90  235 

10 

60 

1 

9.77  963 

9.87  738 

0.12  262 

9.90  225 

9 

10 

9 

59 

2 

9.77  980 

9.87  764 

0.12  236 

9.90  216 

58 

3 

9.77  997 

9.87  790 

0.12  210 

9.90  206 

57 

4 

9.78  013 

9.87  817 

0.12  183 

9.90  197 

56 

17 

26 

10 

5 

9.78  030 

17 
16 
17 
17 

9.87  843 

26 
26 

27 
26 

0.12  157 

9.90  187 

9 
10 

9 
10 

55 

6 

9.78  047 

9.87  869 

0.12  131 

9.90  178 

54 

7 

9.78  063 

9.87  895 

0.12  105 

9.90  168 

53 

8 

9.78  080 

9.87  922 

0.12  078 

9.90  159 

52 

9 

9.78  097 

9.87  948 

0.12  052 

9.90  149 

51 

27  26 

16 

26 

10 

io 

9.78  113 

17 
17 
16 
17 

9.87  974 

26 
27 
26 
26 

0.12  026 

9.90  139 

9 
10 

9 
10 

50 

1 

2.7  2.6 

11 

9.78  130 

9.88  000 

0.12  000 

9.90  130 

49 

2 

5.4  5.2 

12 

9.78  147 

9.88  027 

0.11  973 

9.90  120 

48 

3 

8.1  7.8 

13 

9.78  163 

9.88  053 

0.11  947 

9.90  111 

47 

4 

10.8  10.4 

14 

9.78  180 

9.88  079 

0.11  921 

9.90  101 

46 

5 

13.5  13.0 

17 

26 

10 

6 

16.2  15.6 

15 

9.78  197 

9.88  105 

26 

27 
26 
26 

0.11  895 

9.90  091 

9 
10 

9 
10 

45 

7 

18.9  18.2 

16 

9.78  213 

16 

9.88  131 

0.11  869 

9.90  082 

44 

8 

21.6  20.8 

17 

9.78  230 

17 

9.88, 158 

0.11  842 

9.90  072 

43 

9 

24.3  23.4 

18 

9.78  246 

16 
17 

9.88  184 

0.11  816 

9.90  063 

42 

19 

9.78  263 

9.88  210 

0.11  790 

9.90  053 

41 

17 

26 

10 

20 

9.78  280 

16 
17 
16 
17 

9.88  236 

26 

27 
26 
26 

0.11  764 

9.90  043 

9 
10 
10 

9 

40 

21 

9.78  296 

9.88  262 

0.11  738 

9.90  034 

39 

22 

9.78  313 

9.88  289 

0.11  711 

9.90  024 

38 

23 

9.78  329 

9.88  315 

0.11  685 

9.90  014 

37 

24 

9.78  346 

9.88  341 

0.11  659 

9.90  005 

36 

16 

26 

10 

25 

9.78  362 

17 
16 
17 
16 

9.88  367 

26 

27 
26 
26 

0.11  633 

9.89  995 

10 

9 

10 

10 

35 

26 

9.78  379 

9.88  393 

0.11  607 

9.89  985 

34 

17  16 

27 

9.78  395 

9.88  420 

0.11  580 

9.89  976 

33 

28 

9.78  412 

9.88  446 

0.11  554 

9.89  966 

32 

1 

1.7  1.6 

29 

9.78  428 

9.88  472 

0.11  528 

9.89  956 

31 

2 

3.4  3.2 

17 

26 

9 

3 

5.1  4.8 

30 

9.78  445 

16 
17 
16 
16 

9.88  498 

26 
26 
27 
26 

0.11  502 

9.89  947 

10 

10 

9 

10 

30 

4 

6.8  6.4 

31 

9.78  461 

9.88  524 

0.11  476 

9.89  937 

29 

5 

8.5  8.0 

32 

9.78  478 

9.88  550 

0.11  450 

9.89  927 

28 

6 

10.2  9.6 

33 

9.78  494 

9.88  577 

0.11  423 

9.89  918 

27 

7 

11.9  11.2 

34 

9.78  510 

9.88  603 

0.11  397 

9.89  908 

26 

8 

13.6  12.8 

17 

26 

10 

9 

15.3  14.4 

35 

9.78  527 

16 
17 
16 
16 

9.88  629 

26 
26 
26 
26 

0.11  371 

9.89  898 

10 

9 

10 

10 

25 

36 

9.78  543 

9.88  655 

0.11  345 

9.89  888 

24 

37 

9.78  560 

9.88  681 

0.11  319 

9.89  879 

23 

38 

9.78  576 

9.88  707 

0.11  293 

9.89  869 

22 

39 

9.78  592 

9.88  733 

0.11  267 

9.89  859 

21 

17 

26 

10 

40 

9.78  609 

'  16 
17 
16 
16 

9.88  759 

27 
26 
26 
26 

0.11  241 

9.89  849 

9 
10 
10 
10 

20 

41 

9.78  625 

9.88  786 

0.11  214 

9.89  840 

19 

42 

9.78  642 

9.88  812 

0.11  188 

9.89  830 

18 

43 

9.78  658 

9.88  838 

0.11  162 

9.89  820 

17 

44 

9.78  674 

9.88  864 

0.11  136 

9.89  810 

16 

10   9 

17 

26 

9 

45 

9.78  691 

16 
16 
16 
17 

9.88  890 

26 
26 
26 
26 

0.11  110 

9.89  801 

10 
10 
10 
10 

15 

1 

1.0  0.9 

46 

9.78  707 

9.88  916 

0.11  084 

9.89  791 

14 

2 

2.0  1.8 

47 

9.78  723 

9.88  942 

0.11  058 

9.89  781 

13 

3 

3.0  2.7 

48 

9.78  739 

9.88  968 

0.11  032 

9.89  771 

12 

4 

4.0  3.6 

49 

9.78  756 

9.88  994 

0.11  006 

9.89  761 

11 

5 

5.0  4.5 

16 

26 

9 

6 

6.0  5.4 

50 

9.78  772 

16 
17 
16 
16 

9.89  020 

26 

27 
26 
26 

0.10  980 

9.89  752 

10 
10 
10 
10 

IO 

7 

7.0  6.3 

51 

9.78  788 

9.89  046 

0.10  954 

9.89  742 

9 

8 

8.0  7.2 

52 

9.78  805 

9.89  073 

0.10  927 

9.89  732 

8 

9 

9.0  8.1 

53 

9.78  821 

9.89  099 

0.10  901 

9.89  722 

7 

54 

9.78  837 

9.89  125 

0.10  875 

9.89  712 

6 

16 

26 

10 

55 

9.78  853 

16 
17 
16 
16 

9.89  151 

26 
26 
26 
26 

0.10  849 

9.89  702 

9 
10 
10 
10 

5 

56 

9.78  869 

9.89  177 

0.10  823 

9.89  693 

4 

57 

9.78  S86 

9.89  203 

0.10  797 

9.89  683 

3 

58 

9.78  902 

9.89  229 

0.10  771 

9.89  673 

2 

59 

9.78  918 

9.89  255 

0.10  745 

9.89  663 

1 

16 

26 

10 

60  9.78 

9.89  281 

0.10  719 

9.89  653 

o 

J  LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

/ 

PP 

52°— Logarithms  of  Trigonometric  Functions 


66 


38°—  Logarithms  of  Trigonometric  Functions 


[II 


/ 

LSin 

d 

LTan 

cd 

LCot 

LCos 

d 

PP 

o 

9.78  934 

16 
17 
16 
16 

9.89  281 

26 
26 
26 
26 

0.10  719 

9.89  653 

10 

10 

9 

10 

oo 

1 

9.78  950 

9.89  307 

0.10  093 

9.S9  643 

59 

2 

9.78  967 

9.89  333 

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9.81  417 

9.93  431 

0.06  569 

9.87  985 

19 

42 

9.81  431 

9.93  457 

0.06  543 

9.87  975 

18 

43 

9.81  446 

9.93  482 

0.06  518 

9.87  964 

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44 

9.81  461 

9.93  508 

0.06  492 

9.87  953 

16 

11  10 

14 

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9.81  475 

15 
15 
14 
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9.93  533 

26 
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11 
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1.1  1.0 

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9.81  490 

9.93  559 

0.06  441 

9.87  931 

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2 

2.2  2.0 

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9.81  505 

9.93  584 

0.06  416 

9.87  920 

13 

3 

3.3  3.0 

48 

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9.93  610 

0.06  390 

9.87  909 

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4 

4.4  4.0 

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9.81  534 

9.93  636 

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5 

5.5  5.0 

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6 

6.6  6.0 

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41°— Logarithms  of  Trigonometric  Functions 


69 


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9.82  098 

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9.94  732 

0.05  268 

9.87  423 

28 

6 

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9.82  169 

9.94  757 

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9.87  412 

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10.5  9.8 

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9.82  184 

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9.87  378 

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9.82  226 

9.94  859 

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9.87  367 

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38 

9.82  240 

9.94  884 

0.05  116 

9.87  356 

22 

39 

9.82  255 

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9.82  269 

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14 
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9.94  935 

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9.87  322 

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48°— Logarithms  of  Trigonometric  Functions 


70 


42°— Logarithms  of  Trigonometric  Functions 


1 

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43°— Logarithms  of  Trigonometric  Functions 


71 


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9.83  674 

9.97  523 

0.02  477 

9.86  152 

38 

23 

9.83  688 

9.97  548 

0.02  452 

9.86  140 

37 

• 

24 

9.83  701 

9.97  573 

0.02  427 

9.86  128 

36 

14 

25 

12 

25 

9.83  715 

i3 
13 
14 
13 

9.97  598 

26 
25 
25 
26 

0.02  402 

9.86  116  j  19 
9.86  104   to 

35 

26 

9.83  728 

9.97  624 

0.02  376 

34 

14  IS 

27 

9.83  741 

9.97  649 

0.02  351 

9.86  092 

12 
12 

33 

28 

9.83  755 

9.97  674 

0.02  326 

9.86  080 

32 

1 

1.4  1.3 

29 

9.83  768 

9.97  700 

0.02  300 

9.86  068 

31 

2 

2.8  2.6 

13 

25 

12 

3 

4.2  3.9 

30 

9.83  781 

14 
13 
13 
13 

9.97  725 

25 
26 
25 

25 

0.02  275 

9.86  056 

12 

1  o 

30 

4 

5.6  5.2 

31 

9.83  795 

9.97  750 

0.02  250 

9.86  044 

29 

5 

7.0  6.5 

32 

9.83  808 

9.97  776 

0.02  224 

9.86  032  !  to 
9.86  020   }o 
9.86  008  LZ 

28 

6 

8.4  7.8 

33 

9.83  821 

9.97  801 

0.02  199 

27 

7 

9.8  9.1 

34 

9.83  834 

9.97  826 

0.02  174 

26 

8 

11.2  10.4 

14 

25 

12 

9 

12.6  11.7 

35 

9.83  848 

13 

13 
13 
14 

9.97  851 

26 

25 
25 
26 

0.02  149 

9.85  996  |  19 
9.85  984  1  jo 
9.85  972   jo 
9.85  960  !  to 

25 

36 

9.83  861 

9.97  877 

0.02  123 

24 

37 

9.83  874 

9.97  902 

0.02  098 

23 

38 

9.83  887 

9.97  927 

0.02  073 

22 

39 

9.83  901 

9.97  953 

0.02  047 

9.85  948 

1  — 

21 

13 

25 

12 

40 

9.83  914 

13 
13 
14 
13 

9.97  978 

25 
26 
25 
25 

0.02  022 

9.85  936 

12 
12 
12 

12 

20 

41 

9.83  927 

9.98  003 

0.01  997 

9.85  924 

19 

42 

9.83  940 

9.98  029 

0.01  971 

9.85  912 

18 

43 

9.83  954 

9.98  054 

0.01  946 

9.85  900 

17 

44 

9.83  967 

9.98  079 

0.01  921 

9.85  888 

16 

12  11 

13 

25 

12 

45 

9.83  980 

13 
13 
14 
13 

9.98  104 

26 
25 
25 
26 

0.01  896 

9.85  876 

12 
13 
12 
12 

15 

1 

1.2  1.1 

46 

9.83  993 

9.98  130 

0.01  870 

9.85  864 

14 

2 

2.4  2.2 

47 

9.84  006 

9.98  155 

0.01  845 

9.85  851 

13 

3 

3.6  3.3 

48 

9.84  020 

9.98  180 

0.01  820 

9.85  839 

12 

4 

4.8  4.4 

49 

9.84  033 

9.98  206 

0.01  794 

9.85  827 

11 

5 

6.0  5.5 

13 

25 

12 

6 

7.2  6.6 

50 

9.84  046 

13 
13 
13 
13 

9.98  231 

25 

25 
26 
25 

0.01  769 

9.85  815 

12 
12 
12 
13 

IO 

7 

8.4  7.7 

51 

9.84  059 

9.98  256 

0.01  744 

9.85  803 

9 

8 

9.6  8.8 

52 

9.84  072 

9.98  281 

0.01  719 

9.85  791 

8 

9 

10.8  9.9 

53 

9.84  OSS 

9.98  307 

0.01  693 

9.85  779 

7 

54 

9.84  098 

9.98  332 

0.01  668 

9.85  766 

6 

14 

25 

12 

55 

9.84  112 

13 
13 
13 
13 

9.98  357 

26 
25 
25 
25 

0.01  643 

9.85  754 

12 
12 
12 
12 

5 

56 

9.84  125 

9.98  383 

0.01  617 

9.85  742 

4 

57 

9.84  138 

9.98  408 

0.01  592 

9.85  730 

3 

58 

9.84  151 

9.98  433 

0.01  567 

9.85  718 

2 

59 

9.84  164 

9.98  458 

0.01  542 

9.85  706 

1 

13 

26 

13 

60 

9.84  177 

9.98  484 

O'.Ol  516 

9.85  693 

O 

LCos 

d 

LCot 

cd 

LTan 

LSin  !  d 

t 

PP 

46° — Logarithms  of  Trigonometric  Functions 


72 

44° — Logarithms  of  Trigonometric  Functions 

[II 

r 

LSin 

d 

LTan 

cd 

LCot 

LCos 

d 

PP 

© 

9.84  177 

13 

13 
13 
13 

9.98  484 

25 
25 
26 
25 

0.01  516 

9.85  693 

12 
12 
12 
12 

60 

1 

9.84  190 

9.98  509 

0.01  491 

9.85  681 

59 

2 

9.84  203 

9.98  534 

0.01  466 

9.85  669 

58 

3 

9.84  216 

9.98  560 

0.01  440 

9.85  657 

57 

4 

9.84  229 

9.98  585 

0.01  415 

9.85  645 

56 

13 

25 

13 

5 

9.84  242 

13 
14 
13 
13 

9.98  610 

25 
26 
25 
25 

0.01  390 

9.85  632 

12 
12 
12 
13 

55 

6 

9.84  255 

9.98  635 

0.01  365 

9.85  620 

54 

7 

9.84  269 

9.98  661 

0.01  339 

9.85  608 

53 

8 

9.84  282 

9.98  686 

0.01  314 

9.85  596 

52 

9 

9.84  295 

9.98  711 

0.01  289 

9.85  583 

51 

13 

26 

12 

io 

9.84  308 

13 
13 
13 
13 

9.98  737 

25 
25 
25 
26 

0.01  263 

9.85  571 

12 
12 
13 
12 

50 

11 

9.84  321 

9.98  762 

0.01  238 

9.85  559 

49 

12 

9.84  334 

9.98  787 

0.01  213 

9.85  547 

48 

13 

9.84  347 

9.98  812 

0.01  188 

9.85  534 

47 

14 

9.84  360 

9.98  838 

0.01  162 

9.85  522 

46 

13 

25 

12 

26  25  14 

15 

9.84  373 

12 
13 
13 
13 

9.98  863 

25 

25 
26 
25 

0.01  137 

9.85  510 

13 

12 
12 
13 

45 

16 

9.84  385 

9.98  888 

0.01  112 

9.85  497 

44 

1 

2.6  2.5  1.4 

17 

9.84  398 

9.98  913 

0.01  087 

9.85  485 

43 

2 

5.2  5.0  2.8 

18 

9.84  411 

9.98  939 

0.01  061 

9.85  473 

42 

3 

7.8  7.5  4.2 

19 

9.84  424 

9.98  964 

0.01  036 

9.85  460 

41 

4 

10.4  10.0  5.6 

13 

25 

12 

5 

13.0  12.5  7.0 

20 

9.84  437 

13 
13 
13 
13 

9.98  989 

26 
25 
25 
25 

0.01  Oil 

9.85  448 

12 
13 
12 
12 

40 

6 

15.6  15.0  8.4 

21 

9.84  450 

9.99  015 

0.00  985 

9.85  436 

39 

7 

18.2  17.5  9.8 

22 

9.84  463 

9.99  040 

0.00  960 

9.85  423 

38 

8 

20.8  20.0  11.2 

23 

9.84  476 

9.99  065 

0.00  935 

9.85  411 

37 

9 

23.4  22.5  12.6 

24 

9.84  489 

9.99  090 

0.00  910 

9.85  399 

36 

13 

26 

13 

25 

9.84  502 

13 
13 
12 
13 

9.99  116 

25 
25 
25 
26 

0.00  884 

9.85  386 

12 
13 
12 
12 

35 

26 

9.84  515 

9.99  141 

0.00  859 

9.85  374 

34 

27 

9.84  528 

9.99  166 

0.00  834 

9.85  361 

33 

28 

9.84  540 

9.99  191 

0.00  809 

9.85  349 

32 

29 

9.84  553 

9.99  217 

0.00  783 

9.85  337 

31 

13 

25 

13 

30 

9.84  566 

13 

13 
13 
13 

9.99  242 

25 
26 
25 
25 

0.00  758 

9.85  324 

12 
13 
12 
13 

30 

31 

9.84  579 

9.99  267 

0.00  733 

9.85  312 

29 

32 

9.84  592 

9.99  293 

0.00  707 

9.85  299 

28 

33 

9.84  605 

9.99  318 

0.00  682 

9.85  287 

27 

34 

9.84  618 

9.99  343 

0.00  657 

9.85  274 

26 

12 

25 

12 

35 

9.84  630 

13 
13 
13 
13 

9.99  368 

26 
25 
25 
25 

0.00  632 

9.85  262 

12 
13 
12 
13 

25 

36 

9.84  643 

9.99  394 

0.00  606 

9.85  250 

24 

37 

9.84  656 

9.99  419 

0.00  581 

9.85  237 

23 

38 

9.84  669 

9.99  444 

0.00  556 

9.85  225 

22 

13  12 

39 

9.84  682 

9.99  469 

0.00  531 

9.85  212 

21 

12 

26 

12 

1 

1.3  1.2 

40 

9.84  694 

13 
13 
13 
12 

9.99  495 

25 
25 
25 
26 

0.00  505 

9.85  200 

13 
12 
13 
12 

2© 

2 

2.6  2.4 

41 

9.84  707 

9.99  520 

0.00  480 

9.85  187 

19 

3 

3.9  3.6 

42 

9.84  720 

9.99  545 

0.00  455 

9.85  175 

18 

4 

5.2  4.8 

43 

9.84  733 

9.99  570 

0.00  430 

9.85  162 

17 

5 

6.5  6.0 

44 

9.84  745 

9.99  596 

0.00  404 

9.85  150 

16 

6 

7.8  7.2 

13 

25 

13 

7 

9.1  8.4 

45 

9.84  758 

13 
13 
12 
13 

9.99  621 

25 
26 
25 
25 

0.00  379 

9.85  137 

12 
13 
12 
13 

15 

8 

10.4  9.6 

46 

9.84  771 

9.99  646 

0.00  354 

9.85  125 

14 

9 

11.7  10.8 

47 

9.84  784 

9.99  672 

0.00  328 

9.85  112 

13 

48 

9.84  796 

9.99  697 

0.00  303 

9.85  100 

12 

49 

9.84  809 

9.99  722 

0.00  278 

9.85  087 

11 

13 

25 

13 

5© 

9.84  822 

13 
12 
13 
13 

9.99  747 

26 

25 
25 
25 

0.00  253 

9.85  074 

12 
13 
12 
13 

IO 

51 

9.84  835 

9.99  773 

0.00  227 

9.85  062 

9 

52 

9.84  847 

9.99  798 

0.00  202 

9.85  049 

8 

53 

9.84  860 

9.99  823 

0.00  177 

9.85  037 

7 

54 

9.84  873 

9.99  848 

0.00  152 

9.85  024 

6 

12 

26 

12 

55 

9.84  885 

13 
13 
12 
13 

9.99  874 

25 
25 
25 
26 

0.00  126 

9.85  012 

13 
13 
12 
13 

5 

56 

9.84  898 

9.99  899 

0.00  101 

9.84  999 

4 

57 

9.84  911 

9.99  924 

0.00  076 

9.84  986 

3 

58 

9.84  923 

9.99  949 

0.00  051 

9.84  974 

2 

59 

9.84  936 

9.99  975 

0.00  025 

9.84  961 

1 

13 

25 

12 

60 

9.84  949 

0.00  000 

0.00  000 

9.84  949 

0 

LCos 

d 

LCot 

cd 

LTan 

LSin 

d 

/ 

PP 

45°— Logarithms  of  Trigonometric  Functions 


TABLE  HI 

Values  of  the  Natural  Trigonometric  Functions 

for  Every  Minute  from  0°  to  90° 

to  Five  Decimal  Places 


Ill] 


O6— Natural  Functions— lc 


75 


' 

N  Sin  N  Tan  N  Cot 

NCOS 

0 

.000001.  ooooo  1     oc 

1 . 0000 

«o 

1 

020         020  3437  .  7 

000 

59 

2 

058 

058 

1718.9 

000 

58 

3 

0S7 

087 

1145.9 

000 

57 

4 

116 

116 

859 . 44 

000 

56 

5 

.00145 

.00145 

687. 55  1.0000 

55 

6 

175 

175  572.96 

000 

54 

7 

2041        204  491.11 

000 

53 

8 

233         233  429.72 

000 

52 

9 

262         262  381.97 

000 

51 

io 

.00291  .00291343.77 

1 . 0000 

50 

11 

320 

320  312.52 

.99999 

49 

12 

349 

349  286 .  48 

-999 

48 

13 

378 

378  264.44 

999 

47 

14 

407 

407  245.55 

999 

46 

15 

. 00436 

.00436  229.18 

. 99999 

45 

16 

465 

465  214.86 

999 

44 

17 

495 

495 

202 . 22 

999 

43 

18 

524 

524 

190 . 98 

999 

42 

19 

553 

553 

180.93 

998 

41 

20 

. 00582 

. 00582 

171.89 

. 99998 

40 

21 

611 

611J163.70 

998 

39 

»22 

640 

6401156.26 

998 

38 

23 

669 

669  149.47 

998 

37 

24 

698 

698  143. 24 

998 

36 

25 

. 00727 

.00727  137.51 

.99997 

35 

26 

7561        756  132.22 

997 

34 

27 

785         785J127.32 

997 

33 

28 

814 

815  122.77 

997 

32 

29 

844 

844  118.54 

996 

31 

30 

. 00873 

.00873' 114. 59 

. 99996 

30 

31 

902 

902  110.89 

996 

29 

32 

931 

931  107.43 

996 

28 

33 

960 

960  104.17 

995 

27 

34 

. 00989 

.00989  101.11 

995 

26 

35 

.01018 

.01018'98.218 

. 99995 

25 

36 

047 

047  95 .  489 

995 

24 

37 

076 

076  92.908 

994 

23 

38 

105 

105  90.463 

994 

22 

39 

134 

133  88.144 

994 

21 

40 

.01164 

.01164  85.940 

. 99993 

20 

41 

193 

193  83 .  844 

993 

19 

42 

222 

222  81.847 

993 

18 

43 

251         25179.943 

992 

17 

44 

280.        280  78.126 

992 

16 

45 

.01309'. 01809  76.390 

.99991 

15 

46 

338         338  74.729 

991 

14 

47 

367 

367  73.139 

991 

13 

48 

396 

396  71.615 

990 

12 

49 

425 

425,70.153 

990 

11 

50 

. 01454 

.01455  68.750 

. 99989 

10 

51 

483 

484  67 .  402 

989 

9 

52 

513 

513  66.105 

989 

8 

53 

542 

542  64.858 

988 

7 

54 

571 

571;63.657 

988 

6 

55 

.01600 

.01600  62.499 

. 99987 

5 

56 

629 

629  61.383 

987 

4 

57 

658 

658  60.306 

986 

3 

58 

687 

687159.266 

986 

2 

59 

716 

71658.261 

985 

1 

60 

. 01745 

.01746  57.290 

. 99985 

O 

NCos 

N  Cot  N  Tan 

NSin 

/ 

' 

N  Sin'NTanJN  Cot1  N  Cos 

o 

.01745  .01746  57.290 

.99985 

60 

1 

774|        775  56.351 

984 

59 

2 

8031        804  55.442 

984 

58 

3 

832 

833  54.561 

983 

57 

4 

862 

862  53 .  709 

983 

56 

5 

.01891 

.01891 

52 . 882 

. 99982 

55 

6 

920 

920 

52.081 

982 

54 

7 

949 

949 

51.303 

981 

53 

8 

.01978 

.01978  50.549 

980 

52 

9 

. 02007 

.02007  49.816 

980 

51 

IO 

. 02036 

.  02036149. 104 

. 99979 

50 

11 

065 

066  48.412 

979 

49 

12 

094 

095!  47.  740 

978 

48 

13 

123 

124  47.085 

977 

47 

14 

152 

153 

46 . 449 

977 

46 

15 

.02181 

.02182 

45 . 829 

. 99976 

45 

16 

211!        211 

45 . 226 

976 

44 

17 

240         240 

44 . 639 

975 

43 

18 

269 

269 

44.066 

974 

42 

19 

298 

298 

43 . 508 

974 

41 

20 

. 02327 

. 02328 

42.964 

. 99973 

40 

21 

356 

357 

42 . 433 

972 

39 

22 

385 

386 

41.916 

972 

38 

23 

414 

415  41.41 

971 

37 

24 

443 

444 

40.917 

970 

36 

25 

. 02472 

I 02473 

40 . 436 

. 99969 

35 

26 

501 

502 

39 . 965 

969 

34 

27 

530 

531 

39 . 506 

968 

33 

28 

560 

560 

39 . 057 

967 

32 

29 

589 

589 

38.618 

966 

31 

30 

.02618 

.02619 

38.188 

. 99966 

30 

31 

647 

648 

37 . 769 

965 

29 

32 

676 

677 

37.358 

964 

28 

33 

705 

706 

36 . 956 

963 

27 

34 

734 

735 

36 . 563 

963 

26 

35 

. 02763 

.02764 

36.178 

. 99962 

25 

36 

792 

793 

35.801 

961 

24 

37 

821 

822 

35.431 

960 

23 

38 

850 

851  35.070 

959 

22 

39 

879 

881 

34,715 

959 

21 

40 

. 02908 

.02910 

34.368 

. 99958 

20 

41 

938 

939 

34 . 027 

957 

19 

42 

967 

968 

33 . 694 

956 

18 

43 

. 02996 

.  02997  33 .  366 

955 

17 

44 

.03025 

. 03026 

33.045 

954 

16 

45 

. 03054 

. 03055 

32 . 730 

,99953 

15 

46 

OSS!        084 
ll^        114 

32.421 

952 

14 

47  ' 

32.118 

952 

13 

48 

141 

143 

31.821 

951 

12 

49 

170 

172 

31.528 

950 

11 

50 

.03199 

.03201 

31.242 

. 99949 

IO 

51 

228 

230 

30 . 960 

948 

9 

52 

257 

259 

30.683 

947 

8 

53 

286 

288 

30.412 

946 

7 

54 

316 

317 

30.145 

945 

6 

55 

. 03345 

.03346 

29.882 

. 99944 

5 

56 

374 

376 

29 . 624 

943 

4 

57 

403 

405 

29.371 

942 

3 

58 

432 

434 

29.122 

941 

2 

59 

461 

463 

28.877 

940 

1 

60 

. 03490 

.03492 

28.636 

. 99939 

O 

NCOS 

N  Cot 

NTan 

NSin 

i 

89°— Natural  Functions— 88c 


76 


2°— Natural  Functions— 3C 


[III 


/ 

N  Sin 

NTan 

N  Cot 

N  Cos 

o 

.03490 

. 03492 

28.636 

.99939 

GO 

1 

519 

521 

.399 

938 

59 

2 

548 

550  28.166 

937 

58 

3 

577 

579  27 .  937 

936 

57 

4 

606 

609 

.712 

935 

56 

5 

. 03635 

.03638 

27 . 490 

. 99934 

55 

6 

664 

667 

.271 

933 

54 

7 

693 

696 

27.057 

932 

53 

8 

723 

725 

26.845 

931 

52 

9 

752 

754 

.637 

930 

51 

io 

.03781 

. 03783 

26.432 

. 99929 

50 

11 

810 

812 

.230 

927 

49 

12 

839 

842 

26.031 

926 

48 

13 

868 

871 

25 . 835 

925 

47 

14 

897 

900 

.642 

924 

46 

15 

. 03926 

. 03929 

25.452 

. 99923 

45 

16 

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/ 

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87°— Natural  Functions— 86° 


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4°— Natural  Functions— 5° 


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' 

85°— Natural  Functions— 84c 


6°— Natural  Functions— 7° 


[III 


' 

N  Sin  NTan 

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' 

83°—  Natural  Functions— 82c 


Ill] 


8°— Natural  Functions— 9° 


79 


/ 

N  Sin  N  Tan  N  Cot  N  Cos 

t 

N  SinNTan!N  Cot  N  Cos 

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.14061  .14202 

7.0410  .99006 

55 

5 

. 15787 

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6 . 2549 

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55 

6 

090!   232 

.0264. 99002 

54 

6 

816 

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54 

7 

119 

262 

7.0117  .98998 

53 

7 

845 

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53 

8 

148 

291 

6.9972 

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52 

8 

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52 

9 

177 

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51 

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107 

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51 

10 

. 14205 

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6.9682 

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IO 

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6.1970 

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11 

234 

381 

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11 

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167 

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49 

12 

263 

410 

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48 

12 

. 15988 

196 

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48 

13 

292 

440 

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13 

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47 

14 

320 

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256 

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46 

15 

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6. 8969  .98965 

45 

15 

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6.1402 

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16 

378 

529 

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44 

16 

103 

316 

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44 

17 

407 

559 

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957 

43 

17 

132 

346 

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43 

18 

436 

588 

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953 

42 

18 

160 

376 

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42 

19 

464 

618 

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405 

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6.8269  .98944 

40 

20 

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6 . 0844 

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40 

21 

522 

678 

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39 

21 

246 

465 

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39 

22 

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22 

275 

495 

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38 

23 

580 

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28 

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30 

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31 

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31 

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29 

32 

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28 

32 

562 

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27 

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37 

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5 . 8708 

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19 

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302 

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332 

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6.4348 

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51 

085    570  .4225 

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9 

51 

107 

363 

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52 

414    600  .4103 

805 

8 

52 

136 

393 

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53 

442    630 

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7 

53 

164 

423 

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54 

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6 

54 

193 

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6 

55 

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6.3737 

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5 

55 

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5.7199 

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5 

56 

529 

719 

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4 

56 

250    513 

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501 

4 

57 

557 

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3 

57 

279 

543 

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3 

58 

586 

779 

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2 

58 

308 

573 

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2 

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1 

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60 

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O 

60 

. 17365 

. 17633 

5.6713 

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NCosN  CotNTan  N  Sin 

r 

NCOS 

N  Cot 

NTan 

NSin 

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81°  —Natural  Functions— 80c 


80 


10°— Natural  Functions— llc 


' 

N  Sin 

NTanN  Cot 

N  Cos 

o 

. 17365 

.17633  5.6713 

.98481 

60 

1 

393 

663 

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59 

2 

422 

693 

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58 

3 

451 

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479 

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56 

5 

. 17508 

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5.6234 

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55 

6 

537 

813 

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450 

54 

7 

565 

843 

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53 

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594 

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52 

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623 

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51 

io 

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5 . 5764 

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50 

11 

680 

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49 

12 

708 

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48 

13 

737 

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47 

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15 

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5.5301 

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16 

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44 

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23 

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353 

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25 

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26 

109 

414 

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34 

27 

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33 

28 

166 

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32 

29 

195 

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31 

30 

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5.3955 

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31 

252 

564 

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29 

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281 

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28 

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309 

624 

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35 

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5.3521 

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25 

36 

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41 

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567 

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5.2672 

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15 

46 

681 

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240 

14 

47 

710 

046 

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13 

48 

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50 

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5.2257 

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51 

824 

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53 

881 

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7 

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257 

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6 

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5.1848 

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56 

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317 

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4 

57 

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3 

58 

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378 

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2 

59 

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408 

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1 

60 

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5.1446 

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O 

NCOS 

N  Cot  N  Tan 

NSin 

/ 

o 

1 

2 
3 

4 

5 

6 

7 
8 
9 

IO 

11 
12 
13 
14 

15 

16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 
48 
49 

50 

51 
52 
53 
54 

55 

56 
57 
58 
59 

60 


N  Sin 


19081 
109 
138 
167 
195 

19224 
252 
281 
309 
338 

19366 
395 
423 
452 
481 

19509 
538 
566 
595 
623 

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680 
709 
737 
766 

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880 
908 

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965 

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051 

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108 
136 
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193 

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250 
279 
307 
336 

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393 
421 
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478 

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535 
563 
592 
620 

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677 
706 
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763 

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NTanN  Cot 


NCos 


19438 
468 
498 
529 
559 

19589 
619 
649 
680 
710 

19740 
770 
801 
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861 

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921 

952 

19982 

20012 

20042 
073 
103 
133 
164 

20194 
224 
254 
285 
315 

20345 
376 
406 
436 
466 

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527 
557 
588 
618 

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770 

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043 

073 


5.1446 
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5.1049 
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5 . 0658 
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5.0273 

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0121 

5 . 0045 

4.9969 

4.9894 
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4.9520 
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4.9152 
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4.8788 
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4 . 8430 
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4.8077 
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4.7729 
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N  Cos 


21104  4.7385 


134 
164 
195 
225 

21256 


.7317 
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4.7046 


N  Cot  N  Tan 


98163 
157 
152 
146 
140 

98135 
129 
124 
118 
112 

98107 
101 
096 
090 
084 

98079 
073 
067 
061 
056 

98050 
044 
039 
033 
027 

98021 

016 

010 

98004 

97998 

97992 
987 
981 
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969 

97963 
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952 
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940 

. 97934 
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910 

. 97905 
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893 

887 
881 

.97875 
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863 
857 
851 

.97845 
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833 
827 
821 

.97815 


NSin 


79°— Natural  Functions— 78c 


Ill] 


12°— Natural  Functions— 13° 


t 

N  Sin  N  Tan  N  Cot 

NCOS 

© 

.20791 

.21256 

4.7046 

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1 

820 

286 

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59 

2 

848 

316 

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58 

3 

877 

347 

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57 

4 

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6 

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7 

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53 

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4 . 6382 

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15 

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4.6057 

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44 

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4.5420 

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34 

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29 

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30 

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200 

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261 

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292 

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4.4799 

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25 

36 

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353 

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24 

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383 

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23 

38 

871 

414 

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22 

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444 

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21 

40 

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20 

41 

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560 

19 

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18 

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4.4194 

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15 

46 

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658 

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528 

14 

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13 

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155 

719 

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515 

12 

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183 

750 

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11 

50 

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4.3897 

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811 

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9 

52 

268 

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8 

53 

297 

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325 

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6 

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O 

NCosN  Cot  N  Tan  N  Sin 

/ 

' 

N  Sin  N  Tan  N  Cot  N  Cos 

O 

.22495 

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4.3315 

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117 

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430 

59 

2 

552 

148 

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424 

58 

3 

580 

179 

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57 

4 

608 

209 

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411 

56 

5 

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4.3029 

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55 

6 

665 

271 

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54 

7 

693 

301 

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53 

8 

722 

332 

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52 

9 

750 

363 

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378 

51 

10 

. 22778 

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4.2747 

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50 

11 

807 

424 

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365 

49 

12 

835 

455 

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358 

48 

13 

863 

485 

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351 

47 

14 

892 

516 

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34g 

46 

15 

. 22920 

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4.2468 

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45 

16 

948 

578 

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331 

44 

17 

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608 

.2358 

325 

43 

18 

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639 

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318 

42 

19 

033 

670 

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311 

41 

20 

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4.2193 

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40 

21 

090 

731 

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298 

39 

22 

118 

762 

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291 

38 

23 

146 

793 

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284 

37 

24 

175 

823 

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278 

36 

25 

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4.1922 

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35 

26 

231 

885 

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264 

34 

27 

260 

916 

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257 

33 

28 

288 

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32 

29 

316 

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244 

31 

30 

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4.1653 

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30 

31 

373 

039 

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230 

29 

32 

401 

069 

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223 

28 

33 

429 

100 

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217 

27 

34 

458 

131 

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210 

26 

35 

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4.1388 

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25 

36 

514 

193 

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196 

24 

37 

542 

223 

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189 

23 

38 

571 

254 

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182 

22 

39 

599 

285 

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176 

21 

40 

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4.1126 

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20 

41 

656 

347 

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162 

19 

42 

684 

377 

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155 

18 

43 

712 

408 

.0970 

148 

17 

44 

740 

439 

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141 

16 

45 

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4.0867 

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15 

46 

797 

501 

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127 

14 

47 

825 

532 

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120 

13 

48 

853 

562 

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113 

12 

49 

882 

593 

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106 

11 

50 

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4.0611 

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51 

938 

655 

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9 

52 

966 

686 

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086 

8 

53 

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717 

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7 

54 

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747 

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072 

6 

55 

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4.0358 

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5 

56 

079 

809 

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058 

4 

57 

108 

840 

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051 

3 

58 

136 

871 

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2 

59 

164 

902 

.0158 

037 

1 

60 

.24192 

.24933 

4.0108 

.97030 

O 

NCOS 

N  Cot 

NTan 

NSin 

i 

77°— Natural  Functions— 76° 


82 


14°— Natural  Functions— 15° 


[III 


/ 

N  Sin 

NTanN  Cot 

N  Cos 

o 

.24192 

. 24933 

4.0108 

. 97030 

oo 

1 

220 

964 

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023 

59 

2 

249 

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4.0009 

015 

58 

3 

277 

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3 . 9959 

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57 

4 

305 

056 

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56 

5 

.24333 

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3.9861  .96994 

55 

6 

362 

118 

.9812 

987 

54 

7 

390 

149 

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980 

53 

8 

418 

180 

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973 

52 

9 

446 

211 

.9665 

966 

51 

io 

.24474 

. 25242 

3.9617 

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50 

11 

503 

273 

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49 

12 

531 

304 

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13 

559 

335 

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47 

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587 

366 

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46 

15 

. 24615 

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3 . 9375 

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45 

16 

644 

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43 

18 

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19 

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41 

20 

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3.9136 

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40 

21 

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22 

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614 

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30 

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3 . 8667 

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31 

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35 

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3 . 8436 

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25 

36 

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048 

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24 

37 

235 

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38 

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110 

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21 

40 

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3 . 8208 

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20 

41 

348 

203 

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19 

42 

376 

235 

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18 

43 

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266 

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17 

44 

432 

297 

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712 

16 

45 

. 25460 

. 26328 

3.7983 

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15 

46 

488 

359 

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14 

47 

516 

390 

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13 

48 

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421 

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12 

49 

573 

452 

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11 

50 

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3 .  7760 

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51 

629 

515 

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9 

52 

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546 

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8 

53 

685 

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7 

54 

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6 

55 

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3.7539 

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5 

56 

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4 

57 

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701 

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3 

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826 

733, 

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608 

2 

59 

854 

764 

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1 

60 

.25882 

. 26795 

3.7321 

. 96593 

O 

NCos 

N  Cot 

NTan 

NSin 

' 

/ 

N  Sin  NTanN  Cot1  N  Cos 

o 

.25882 

.26795  3.7321 

. 96593 

60 

1 

910 

826 

.7277 

585 

59 

2 

938 

857 

.7234 

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58 

3 

966 

888 

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570 

57 

4 

. 25994 

920 

.7148 

562 

56 

5 

. 26022 

.26951 

3.7105 

. 96555 

55 

6 

050  .26982 

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547 

54 

7 

079  .27013 

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53 

8 

107 

044 

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52 

9 

135 

076 

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51 

IO 

.26163 

.27107 

3.6891 

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50 

11 

191 

138 

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49 

12 

219 

169 

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48 

13 

247 

201 

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47 

14 

275 

232 

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46 

15 

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. 27263 

3.6680 

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45 

16 

331 

294 

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44 

17 

359 

326 

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43 

18 

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3 . 6470 

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21 

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39 

22 

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23 

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3 . 6264 

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34 

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3 . 6059 

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30 

31 

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29 

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26 

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3 . 5856 

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25 

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24 

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21 

40 

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3 . 5656 

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20 

41 

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19 

42 

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18 

43 

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17 

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16 

45 

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3 . 5457 

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15 

46 

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14 

47 

200 

266 

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230 

13 

48 

228 

297 

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12 

49 

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329 

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214 

11 

50 

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3.5261 

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51 

312 

391 

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198 

9 

52 

340 
368 

423 

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190 

8 

53 

454 

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182 

7 

54 

396 

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6 

55 

. 27424 

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3 . 5067 

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5 

56 

452 

549 

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4 

57 

480 

580 

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3 

58 

508 

612 

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142 

2 

59 

536 

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134 

1 

OO 

.27564 

. 28675 

3.4874 

.96126 

O 

NCOS 

N  Cot 

NTan 

NSin 

f 

75°— Natural  Functions— 74c 


16°— Natural  Functions— 17c 


83 


/ 

N  Sin  N  Tan  N  Cot 

NCOS 

o 

.27564  .28675  3.4874 

.96126 

60 

1 

592    706 

.4836 

■  118 

59 

2 

620    738 

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110 

58 

3 

648    769 

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57 

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5 

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55 

6 

731 

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54 

7 

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895 

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53 

8 

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52 

9 

815 

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51 

io 

. 27843 

. 28990 

3.4495 

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50 

11 

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49 

12 

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48 

13 

927 

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47 

14 

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116 

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46 

15 

. 27983 

. 29147 

3 . 4308 

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45 

16 

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179 

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44 

17 

039 

210 

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43 

18 

067    242 

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42 

19 

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274 

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41 

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3.4124 

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40 

21 

150 

337 

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39 

22 

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368 

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38 

23 

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400 

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37 

24 

234 

432 

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36 

25 

. 28262 

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3.3941 

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35 

26 

290 

495 

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915 

34 

27 

318 

526 

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33 

28 

346 

558 

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32 

29 

374 

590 

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31 

30 

. 28402 

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3.3759 

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30 

31 

429 

653 

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874 

-29 

32 

457 

685 

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28 

33 

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27 

34 

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26 

35 

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3 . 3580 

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25 

36 

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24 

37 

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843 

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23 

38 

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22 

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652 

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21 

40 

. 28680 

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3 . 3402 

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20 

41 

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791 

19 

42 

736 

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18 

43 

764 

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17 

44 

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065 

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16 

45 

. 28820 

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3.3226 

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15 

46 

847 

128 

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749 

14 

47 

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160 

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740 

13 

48 

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192 

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12 

49 

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11 

50 

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. 30255 

3 . 3052 

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IO 

51 

. 28987 

287 

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9 

52 

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319 

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8 

53 

042 

351 

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7 

54 

070 

382 

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681 

6 

55 

. 29098 

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3.2879 

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5 

56 

126 

446 

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4 

57 

154 

478 

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656 

3 

58 

182 

509 

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2 

59 

209 

541 

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639 

1 

60 

.29237 

. 30573 

3 . 2709 

. 95630 

O 

N  Cos  N  Cot  N  Tan  N  Sin 

/ 

/ 

N  Sin 

NTan 

N  Cot 

N  Cos 

60 

O 

.  29237 

. 30573 

3 . 2709 

. 95630 

1 

265 

605 

.2675 

622 

59 

2 

293 

637 

.2641 

613 

58 

3 

321 

669 

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57 

4 

348 

700 

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56 

5 

. 29376 

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3 . 2539 

. 95588 

55 

6 

404 

764 

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579 

54 

7 

432 

796 

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53 

8 

460 

828 

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562 

52 

9 

487 

860 

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51 

IO 

.29515 

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3.2371 

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50 

11 

543 

923 

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536 

49 

12 

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955 

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48 

13 

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47 

14 

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46 

15 

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3 . 2205 

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45 

16 

682 

083 

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493 

44 

17 

710 

115 

.2139 

485 

43 

18 

737 

147 

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476 

42 

19 

765 

178 

.2073 

467 

41 

20 

. 29793 

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3.2041 

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40 

21 

821 

242 

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450 

39 

22 

849 

274 

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441 

38 

23 

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306 

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433 

37 

24 

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36 

25 

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3 . 1878 

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35 

26 

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402 

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407 

34 

27 

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434 

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398 

33 

28 

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466 

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32 

29 

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498 

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380 

31 

30 

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.31530 

3.1716 

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30 

31 

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562 

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363 

29 

32 

126 

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354 

28 

33 

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626 

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27 

34 

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337 

26 

35 

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3.1556 

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25 

36 

237 

722 

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319 

24 

37 

265 

754 

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310 

23 

38 

292 

786 

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301 

22 

39 

320 

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293 

21 

40 

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3.1397 

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20 

41 

376 

882 

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275 

19 

42 

403 

914 

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266 

18 

43 

431 

946 

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257 

17 

44 

459 

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^271 

248 

16 

45 

. 30486 

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15 

46 

514 

042 

.1209 

231 

14 

47 

542 

074 

.1178 

222 

13 

48 

570 

106 

.1146 

213 

12 

49 

597 

139 

.1115 

204 

11 

50 

.30625 

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3 . 1084 

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IO 

51 

653 

203 

.1053 

186 

9 

52 

680 

235 

.1022 

177 

8 

53 

708 

267 

.0991 

168 

7 

54 

736 

299 

.0961 

159 

6 

55 

. 30763 

.32331  3.0930 

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5 

56 

791 

363 

.0899 

142 

4 

57 

819 

396 

.0868 

133 

3 

58 

846 

428 

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124 

2 

59 

874 

460 

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115 

1 

60 

. 30902 

NCos 

. 32492 

3 . 0777 

.95106 

O 

N  Cot 

NTan 

NSin 

/ 

73°— Natural  Functions— 72c 


84 


18°— Natural  Functions— 19° 


/ 

N  SinNTanN  Cot  N  Cos 

/ 

N  Sin 

NTan 

N  Cot 

N  Cos 

o 

. 30902 

.32492 

3.0777 

. 95106 

60 

o 

. 32557 

. 34433 

2.9042 

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oo 

1 

929 

524 

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097 

59 

1 

584 

465 

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542 

59 

2 

957 

556 

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088 

58 

2 

612 

498 

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533 

58 

3 

. 30985 

588 

.0686 

079 

57 

3 

639 

530 

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523 

57 

4 

.31012 

621 

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070 

56 

4 

667 

563 

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514 

56 

5 

.31040 

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3 . 0625 

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55 

5 

. 32694 

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2.8905 

. 94504 

55 

6 

068 

685 

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052 

44 

6 

722 

628 

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495 

54 

7 

095 

717 

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043 

53 

7 

749 

661 

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485 

53 

8 

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749 

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033 

52 

8 

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693 

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52 

9 

151 

782 

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51 

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720 

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3.0475 

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50 

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2.8770 

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50 

11 

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49 

11 

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791 

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49 

12 

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878 

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12 

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824 

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48 

13 

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14 

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943 

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46 

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3 . 0326 

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45 

15 

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2.8636 

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45 

16 

344 

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44 

16 

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954 

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399 

44 

17 

372 

040 

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952 

43 

17 

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43 

18 

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072 

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18 

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42 

19 

427 

104 

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41 

19 

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052 

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41 

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3.0178 

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40 

20 

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2 . 8502 

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40 

21 

482 

169 

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915 

39 

21 

134 

118 

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351 

39 

22 

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201 

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38 

22 

161 

150 

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342 

38 

23 

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233 

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37 

23 

189 

183 

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35 

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2.8370 

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3 . 0003 

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34 

26 

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27 

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363 

2 . 9974 

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33 

27 

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314 

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293 

33 

28 

675 

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32 

28 

326 

346 

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284 

32 

29 

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31 

29 

353 

379 

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31 

30 

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2 . 9887 

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30 

30 

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2 . 8239 

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30 

31 

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492 

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29 

31 

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445 

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29 

32 

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814 

28 

32 

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477 

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28 

33 

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557 

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27 

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510 

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27 

34 

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26 

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2.9743 

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25 

35 

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2.8109 

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25 

36 

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654 

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24 

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24 

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23 

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23 

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22 

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21 

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2 .  9600 

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20 

40 

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2 . 7980 

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20 

41 

034 

816 

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19 

41 

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157 

19 

42 

061 

848 

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721 

18 

42 

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147 

18 

43 

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881 

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712 

17 

43 

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838 

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137 

17 

44 

116 

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16 

44 

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127 

16 

45 

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2 . 9459 

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15 

45 

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2 . 7852 

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15 

46 

171 

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684 

14 

46 

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108 

14 

47 

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13 

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13 

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227 

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12 

48 

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12 

49 

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11 

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035 

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11 

50 

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2.9319 

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50 

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2.7725 

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IO 

51 

309 

140 

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637 

9 

51 

956 

101 

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9 

52 

337 

173 

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627 

8 

52 

. 33983 

134 

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049 

8 

53 

364 

205 

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618 

7 

53 

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167 

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039 

7 

54 

392 

238 

.9208 

609 

6 

54 

038 

199 

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029 

6 

55 

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2.9180 

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5 

55 

. 34065 

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2 . 7600 

.94019 

5 

56 

447 

303 

.9152 

590 

4 

56 

093 

265 

.  7575 

. 94009 

4 

57 

474 

335 

.9125 

580 

3 

57 

120 

298 

.7550 

.03999 

3 

58 

502 

368 

.901)7 

571 

2 

58 

147 

331 

.7525 

989 

2 

59 

529 

400 

.9070 

561 

1 

59 

175 

364 

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979 

1 

eo 

. 32557 

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2.9042 

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O 

60 

.34202 

. 36397 

2.7475 

. 93969 

O 

N  Cos  N  Cot 

NTan 

NSin 

' 

N  Cos  N  Cot 

NTan 

NSin 

71°— Natural  Functions— 70c 


20°— Natural  Functions— 21c 


85 


/ 

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1 

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N  Cot 

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/ 

/ 

N  Sin 

NTan 

N  Cot 

NCos 

o 

. 35837 

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2.6051 

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NCOS 

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t 

69°— Natural  Functions— 68c 


86 


22°— Natural  Functions— 23c 


' 

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o 

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2.4751  .92718 

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/ 

N  Sin  N  Tan  N  Cot|  N  Cos 

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N  Cos  N  Cot 

NTan 

NSin 

/ 

67°— Natural  Functions— 66c 


Ill] 


24°— Natural  Functions— 25c 


87 


/ 

N  Sin  N  Tan  N  Cot  N  Cos 

o 

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2.2460 

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' 

/ 

N  Sin1  N  Tan  N  Cot 

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o 

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2 

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1 

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O 

NCOS 

N  Cot  N  Tan 

NSin 

/ 

65°— Natural  Functions— 64c 


88 


26°— Natural  Functions— 2T 


.[III 


' 

N  Sin  N  Tan  N  Cot 

NCos 

o 

. 43837 

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NCOS 

N  Cot 

NTan 

NSin 

/ 

/ 

N  Sin  NTan 

N  Cot 

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o 

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NCOS 

N  Cot 

NTan 

NSin 

63°— Natural  Functions— 62c 


28°— Natural  Functions— 29° 


89 


/ 

N  Sin 

NTanjN  Cot  N  Cos 

o 

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52 

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N  Cos  N  Cot 

N  Tan  N  Sin 

' 

61°— Natural  Functions— 60c 


90 


30°— Natural  Functions— 31/ 


1 

N  Sin  N  Tan  N  Cot  N  Cos 

o 

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.  6066 

897 

6 

55 

52869 

.62283 

1 . 6055 

. 84882 

5 

56 

893    325 

.6045 

866 

4 

57 

918 

366 

.6034 

3 

58 

943 

406 

.  6024 

S36 

2 

59 

967 

446 

.6014 

820 

1 

60 

. 52992 

.62487 

1  .•;<  103  .84805 

0 

L  .,._ 

NCos 

N  Cot 

NTan  N  Sin 

/ 

59°— Natural  Functions— 58c 


32°— Natural  Functions— 33c 


91 


/ 

N  Sin  N  Tan  N  Cot1  N  Cos 

o 

. 52992 

.  62487  1 .  6003 

.84805 

60 

1 

.53017    527 

.5993    789 

59 

2 

041    568 

.5983    774 

58 

3 

066 

608 

.5972    759 

57 

4 

091 

649 

.5962 

743 

56 

5 

.53115 

. 62689 

1 . 5952 

. 84728 

55 

6 

140 

730 

.5941 

712 

54 

7 

164 

770 

.5931 

697 

53 

8 

189 

811 

.5921 

681 

52 

9 

214 

852 

.5911 

666 

51 

10 

. 53238 

. 62892 

1 . 5900 

.84650 

50 

11 

263 i   933 

.5890    635 

49 

12 

288  .62973 

.5880    619 

48 

13 

312  .63014 

.5869    604 

47 

14 

337 

055 

.5859    588 

46 

15 

.53361 

. 63095 

1.5849  .84573 

45 

16 

386 

136 

.5839    557 

44 

17 

411 

177 

.5829    542 

43 

18 

435 

217 

.5818!   526 

42 

19 

460 

258 

.5808    511 

41 

20 

. 53484 

. 63299 

1.5798  .84495 

40 

21 

509 

340 

.5788    480 

39 

22 

534 

380 

.5778    464 

38 

23 

558 

421 

.5768    448 

37 

24 

583 

462 

.5757    433 

36 

25 

. 53607 

. 63503 

1.5747  .84417 

35 

26 

632 

544 

.5737    402 

34 

27 

656 

584 

.5727    386 

33 

28 

681 

625 

.5717    370 

32 

29 

705 

666 

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31 

30 

.53730  .63707 

1.5697  .84339 

30 

31 

754 

748 

. 56S7    324 

29 

32 

779 

789 

.  5677    308 

28 

33 

804 

830 

.5667    292 

27 

34 

828 

871 

.5657,   277 

26 

35 

. 53853 

.63912 

1.5647  .84261 

25 

36 

877 

953 

.  5637    245 

24 

37 

902  .63994 

.5627    230 

23 

38 

926  .64035 

.5617    214 

22 

39 

951    076 

.5607,    198 

21 

40 

.53975  .64117 

1.5597  .84182 

20 

41 

. 54000    158 

.  5587    167 

19 

42 

024    199 

.5577    151 

18 

43 

049    240 

.5567    135 

17 

44 

073    281 

.5557    120 

16 

45 

.54097  .64322 

1.5547  .84104 

15 

46 

122    363 

.  5537    088 

14 

47 

146!   404 

.  5527    072 

13 

48 

171    446 

.5517    057 

12 

49 

195.   487 

.5507    041 

11 

50 

.54220  .64528 

1.5497  .84025 

IO 

51 

244    56ft 

tfK  84009 

9 

52 

269    6»    H  83994 

8 

53 

293    eoS^WKs*   978 

7 

54 

317    693  .54158    962 

6 

55 

.54342  .64734  1.5448  .83946 

5 

56 

3661   775  .5438    930 

4 

57 

391.    817 

.5428    915 

3 

58 

415 

858 

.5418    899 

2 

59 

440 

899 

.  6408    883 

1 

OO 

.54464 

.64941 

1.5399  .83867 

1 

O 

N  Cos  N  Cot  N  Tan  N  Sin 

' 

f 

N  SinNTanN  Cot  N  Cos 

O 

.54464  .64941  1.5399  .83867 

60 

1 

488i.64982|  .5389 

851 

59 

2 

513 

. 65024 !  .5379 

835 

58 

3 

537 

O65  .5369 

819 

57 

4 

561 

106 

.5359 

804 

56 

5 

.54586 

. 65148 

1.5350  .83788 

55 

6 

610 

189 

.5340    772 

54 

7 

635 

231 

.5330    756 

53 

8 

659 

272 

.5320    740 

52 

9 

683 

314 

.5311    724 

51 

10 

.54708 

. 65355 

1.5301  .83708 

50 

11 

732 

397  .5291'   692 

49 

12 

756 

438 

.5282:   676 

48 

13 

781 

480 

.5272    660 

47 

14 

805 

521 

.5262j   645 

46 

15 

.  54829 

. 65563 

1.5253  .83629 

45 

16 

854 

604(  .5243!   613 

44 

17 

878 

646 

.5233    597 

43 

18 

902 

688 

.5224    581 

42 

19 

927 

729 

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41 

20 

.54951 

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1.5204  '.83549 

40 

21 

975    813 

.5195    533 

39 

22 

.54999!   854 

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38 

23 

. 55024 i   896 

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37 

24 

048    93S 

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36 

25 

.55072 '.65980 

1.5156  .83469 

35 

26 

097  .66021 

.5147    453 

34 

27 

121    063 

.5137    437 

33 

23 

145 

105 

.5127    421 

32 

29 

169 

147 

.5118!   405 

31 

30 

.55194 

.66189 

1.5108  .83389 

30 

31 

218    230 

.5099!   373 

29 

32 

242    272 

.  5089    356 

28 

33 

266    314 

.5080    340 

27 

34 

291 r   356 

.5070    324 

26 

35 

.55315  .66398 

1.5061  .83308 

25 

36 

339;   440 

.5051    292 

24 

37 

363    482 

.5042    276 

23 

38 

388!   524 

. 5032    260 

22 

39 

412 

,   566 

.5023    244 

21 

40 

. 55436 

. 66608 

1.5013  .83228 

20 

41 

460 

650 

.5004    212 

19 

42 

484 

692 

.  4994    195 

18 

43 

509 

734 

.4985    179 

17 

44 

533 

776 

.4975 

163 

16 

45 

.55557  .66818 

1 . 4966 

.83147 

15 

46 

581    860 

.4957 

131 

14 

47 

605!    902 

.4947 

115 

13 

48 

630    944 

.4938 

098 

12 

49 

654,.  66986 

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082 

11 

SO 

.55678 '.67028 

1.4919 

. 83066 

IO 

51 

702    071 

.4910 

050 

9 

52 

72fi    113 

.4900 

034 

8 

53 

750    155 

.4891 

017 

7 

54 

775    197 

.4882 

.83001 

6 

55 

.55799  .67239 

1 . 4872 

. 82985 

5 

56 

823    2S2  •  .4863 

969 

4 

57 

847!   324 

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953 

3 

58 

87 1    366 

. 4844    936 

2 

59 

895    409 

.4835 

920 

1 

60 

.559 19  .67451 

1.4826 

.82904 

O 

NCosN  Cot  N  Tan 

NSin 

/ 

57°— Natural  Functions— 56c 


92 


34°— Natural  Functions— 35c 


/ 

N  Sin 

NTan'N  Cot  N  Cos 

o 

.55919 

.67451 

1.4826  .82904 

60 

1 

943 

493 

.4816!   887 

59 

2 

968 

536 

.48071   871 

58 

3 

. 55992 

578 

.4798    855 

57 

4 

.56016 

620 

.4788 

839 

56 

5 

. 56040 

. 67663 

1.4779 

. 82822 

55 

6 

064 

705 

.4770 

806 

54 

7 

088 

748 

.4761 

790 

53 

8 

112 

790 

.4751 

773 

52 

9 

136 

832 

.4742 

757 

51 

io 

.56160 

.67875 

1 . 4733 

.82741 

50 

11 

184 

917 

.4724 

724 

49 

12 

208 

. 67960 

.4715 

708 

48 

13 

232 

. 68002 

.4705 

692 

47 

14 

256 

045 

.4696 

675 

46 

15 

.  56280 

. 68088 

1.4687 

. 82659 

45 

16 

305 

130 

.4678 

643 

44 

17 

329 

173 

.4669 

626 

43 

18 

353 

215 

.4659 

610 

42 

19 

377 

258 

.4650 

593 

41 

20 

. 56401 

. 68301 

1.4641 

. 82577 

40 

21 

425 

343 

.4632 

561 

39 

22 

449 

386 

.4623 

544 

38 

23 

473 

429 

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528 

37 

24 

497 

471 

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511 

36 

25 

.56521 

. 68514 

1.4596 

. 82495 

35 

26 

545 

557 

.4586 

478 

34 

27 

569 

600 

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462 

33 

28 

593 

642 

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446 

32 

29 

617 

685 

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429 

31 

30 

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. 68728 

1 . 4550 

.82413 

30 

31 

665 

771 

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396 

29 

32 

689 

814 

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380 

28 

33 

713 

857 

.4523 

363 

27 

34 

736 

900 

.4514 

347 

26 

35 

. 56760 

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1.4505 

.82330 

25 

36 

784 

. 68985 

.4496 

314 

24 

37 

808 

. 69028 

.4487 

297 

23 

38 

832 

071 

.4478 

281 

22 

39 

856 

114 

.4469 

264 

21 

40 

. 56880 

.69157 

1.4460 

. 82248 

20 

41 

904 

200 

.4451 

231 

19 

42 

928 

243 

.4442 

214 

18 

43 

952 

286 

.4433 

198 

17 

44 

. 56976 

329 

.4424 

181 

16 

45 

. 57000 

. 69372 

1.4415 

.82165 

15 

46 

024 

416 

.4406 

148 

14 

47 

047 

459 

.4397 

132 

13 

48 

071 

502 

.4388 

115 

12 

49 

095 

545 

.4379 

098 

11 

50 

.57119 

. 69588 

1.4370 

. 82082 

IO 

51 

143 

631 

.4361 

065 

9 

52 

167 

675 

.4352 

048 

8 

53 

191 

718 

.4344 

032 

7 

54 

215 

761 

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6 

55 

. 57238 

.69804 

1.4326 

.81999 

5 

56 

262 

847 

.4317 

982 

4 

57 

286 

891 

.4308 

965 

3 

58 

310 

934 

.4299 

949 

2 

59 

334 

. 69977 

.4290 

932 

1 

60 

.57358 

.70021 

1.4281 

.81915 

O 

NCOS 

N  Cot 

NTan 

NSin 

i 

i 

N  Sin  NTan 

N  Cot 

N  Cos 

o 

. 57358 

.70021 

1.4281 

.81915 

OO 

1 

381 

064 

.4273 

899 

59 

2 

405 

107 

.4264 

882 

58 

3 

429 

151 

.4255 

865 

57 

4 

453 

194 

.4246 

848 

56 

5 

. 57477 

. 70238 

1.4237 

.81832 

55 

6 

501 

281 

.4229 

815 

54 

7 

524 

32g 

.4220 

798 

53 

8 

548 

368 

.4211 

782 

52 

9 

572 

412 

.4202 

765 

51 

IO 

. 57596 

.70455 

1.4193 

.81748 

50 

11 

619 

499 

.4185 

731 

49 

12 

643 

542 

.4176 

714 

48 

13 

667 

586 

.4167 

698 

47 

14 

691 

629 

.4158 

681 

46 

15 

.57715 

. 70673 

1.4150 

. 81664 

45 

16 

738 

717 

.4141 

647 

44 

17 

762 

760 

.4132 

631 

43 

18 

786 

804 

.4124 

614 

42 

19 

810 

848 

.4115 

597 

41 

20 

. 57833 

. 70891 

1 . 4106 

. 81580 

40 

21 

857 

935 

.4097 

563 

39 

22 

881 

. 70979 

.4089 

546 

38 

23 

904 

.71023 

.4080 

530 

37 

24 

928 

066 

.4071 

513 

36 

25 

. 57952 

.71110 

1 . 4063 

. 81496 

35 

26 

976 

154 

.4054 

479 

34 

27 

. 57999 

198 

.4045 

462 

33 

28 

. 58023 

242 

.4037 

445 

32 

29 

047 

285 

.4028 

428 

31 

30 

. 58070 

.71329 

1.4019 

.81412 

30 

31 

094 

373 

.4011 

395 

29 

32 

118 

417 

.4002 

378 

28 

33 

141 

461 

.3994 

361 

27 

34 

165 

505 

.3985 

344 

26 

35 

.58189 

.71549 

1 . 3976 

.81327 

25 

36 

212 

593 

.3968 

310 

24 

37 

236 

637 

.3959 

293 

23 

38 

260 

681 

.3951 

276 

22 

39 

283 

725 

.3942 

259 

21 

40 

. 58307 

.71769 

1 . 3934 

.81242 

20 

41 

330 

813 

.3925 

225 

19 

42 

354 

857 

.3916 

208 

18 

43 

378 

901 

.3908 

191 

17 

44 

401 

946 

.3899 

174 

16 

45 

. 58425 

.71990 

1.3891 

.81157 

15 

46 

449 

.72034 

.3882 

140 

14 

47 

472 

078 

.3874 

123 

13 

48 

496 

122 

.3865 

106 

12 

49 

519 

167 

.3857 

089 

11 

50 

. 58543 

.72211 

1 . 3848 

.81072 

IO 

51 

567 

255 

.3840 

055 

9 

52 

590 

299 

.3831 

038 

8 

53 

614 

344 

.3823 

021 

7 

54 

637 

388 

.3814 

.81004 

6 

55 

.58661 

. 72432 

1 . 3806 

. 80987 

5 

56 

684 

477 

.3798 

970 

4 

57 

708 

521 

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953 

3 

58 

731 

565 

.3781 

936 

2 

59 

755 

610 

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919 

1 

60 

. 58779 

.  72654 

1.3764 

. 80902 

O 

NCos 

N  Cot 

NTan 

NSin 

/ 

55°— Natural  Functions— 54° 


36°— Natural  Functions— 37c 


93 


/ 

N  Sin 

NTan'N  Cot 

N  Cos 

/ 

N  Sin  NTan'N  Cot 

NCos 

o 

. 58779 

. 72654 

1 .  3764 

.80902 

60 

o 

. 60182 

.75355  1.3270 

. 79864 

60 

1 

802 

699 

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885 

59 

1 

205 

401 

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846 

59 

2 

826 

743 

.3747 

867 

58 

2 

228 

447 

.3254 

829 

58 

3 

849 

788 

.3739 

850 

57 

3 

251 

492 

.3246 

811 

57 

4 

873 

832 

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833 

56 

4 

274 

538 

.3238 

793 

56 

5 

.58896 

. 72877 

1.3722 

.80816 

55 

5 

. 60298 

. 75584 

1.3230 

. 79776 

55 

6 

920 

921 

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799 

54 

6 

321 

629 

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758 

54 

7 

943 

. 72966 

.3705 

782 

53 

7 

344 

675 

.3214 

741 

53 

8 

967 

.73010 

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765 

52 

8 

367 

721 

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723 

52 

9 

. 58990 

055 

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748 

51 

9 

390 

767 

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706 

51 

io 

.59014 

.73100 

1.3680 

.80730 

50 

10 

.60414 

.75812 

1.3190 

.79688 

50 

11 

037 

144 

.3672 

713 

49 

11 

437 

858 

.3182 

671 

49 

12 

061 

189 

.3663 

696 

48 

12 

460 

904 

.3175 

653 

48 

13 

084 

234 

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679 

47 

13 

483 

950 

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635 

47 

14 

108 

278 

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662 

46 

14 

506 

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618 

46 

15 

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. 73323 

1.3638 

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45 

15 

. 60529 

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1.3151 

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45 

16 

154 

368 

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627 

44 

16 

553 

088 

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583 

44 

17 

178 

413 

.3622 

610 

43 

17 

576 

134 

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565 

43 

18 

201 

457 

.3613 

593 

42 

18 

599 

180 

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547 

42 

19 

225 

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576 

41 

19 

622 

226 

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530 

41 

20 

. 59248 

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1.3597 

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40 

20 

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1.3111 

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40 

21 

272 

592 

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541 

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318 

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39 

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637 

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38 

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364 

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37 

24 

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36 

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36 

25 

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1.3555 

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35 

25 

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1.3072 

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35 

26 

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34 

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34 

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33 

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33 

28 

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32 

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32 

29 

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951 

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31 

29 

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353 

31 

30 

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1.3514 

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30 

30 

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1 . 3032 

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30 

31 

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29 

31 

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29 

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086 

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26 

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1.3473 

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25 

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1 . 2993 

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25 

36 

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267 

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24 

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16 

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429 

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55 

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1 . 2838 

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4 

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1 . 2799 

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NCosJN  Cot 

NTan 

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' 

NCOS 

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NTan 

NSin 

/ 

53°— Natural  Functions— 52° 


94 


38°— Natural  Functions— 39d 


' 

N  Sin  N  Tan  N  Cot  N  Cos 

o 

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1.2799  .78801 

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NCOS 

N  Cot 

NTan 

NSin 

/ 

51°— Natural  Functions— 50c 


406— Natural  Functions— 41( 


1 

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NTan 

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0 

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NCosN  CotNTan!  N  Sin 

/ 

49°— Natural  Functions— 48c 


96 


426—  Natural  Functions— 43° 


t 

N  Sin 

NTanN  Cot  N  Cos 

N  Sin 

NTan 

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o 

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49 

965 

655 

.0793 

353 

11 

49 

235 

. 95952 

.0422 

156 

11 

50 

. 67987 

. 92709 

1 . 0786 

.73333 

IO 

50 

. 69256 

. 96008 

1.0416 

.72136 

IO 

51 

. 68008 

763 

.0780 

314 

9 

51 

277 

064 

.0410 

116 

9 

52 

029 

817 

.0774 

294 

8 

52 

298 

120 

.0404 

095 

8 

53 

051 

872 

.0768 

274 

7 

53 

319 

176 

.0398 

075 

I 

54 

072 

926 

.0761 

254 

6 

54 

340 

232 

.0392 

055 

6 

55 

. 68093 

. 92980 

1 . 0755 

. 73234 

5 

55 

. 69361 

. 96288 

1.0385 

.72035 

5 

56 

us 

. 93034 

.0749 

215 

4 

56 

382 

344 

.0379 

.72015 

4 

57 

136 

088 

.0742 

195 

3 

57 

403 

400 

.0373 

.71995 

3 

58 

157 

143 

.0736 

175 

2 

58 

424 

457 

.0367 

974 

2 

59 

17S 

197 

.0730 

155 

1 

59 

44S 

513 

.0361 

954 

1 

60 

.6820C 

. 93252 

1 . 0724 

.73135 

O 

60 

. 69466 

. 96569 

1 . 0355 

7 . 1934 

O 

N  Cos  N  Col 

NTan 

NSin 

/ 

NCos 

N  Cot 

NTan 

NSin 

/ 

47°— Natural  Functions— 46° 


44°— Natural  Functions 


97 


' 

N  Sin  N  Tan!  N  Cot  N  Cos 

o 

.69466 

.96569  1.0355 

.71934 

60 

1 

487 

625  .  0349 

914 

59 

2 

508 

681 

.0343 

894 

58 

3 

529 

738 

.0337 

873 

57 

4 

549 

794 

.0331 

853 

56 

5 

. 69570 

. 96850 

1 . 0325 

.71833 

55 

6 

591 

907 

.0319 

813 

54 

7 

612 

. 96963 

.0313 

792 

53 

8 

633 

. 97020 

.0307 

772 

52 

9 

654 

076 

.0301 

752 

51 

io 

.69675 

.97133 

1 . 0295 

.71732 

50 

11 

696 

189 

.0289 

711 

49 

12 

717 

246 

.0283 

691 

48 

13 

737 

302 

.0277 

671 

47 

14 

758 

359 

.0271 

650 

46 

15 

. 69779 

.97416 

1 . 0265 

.71630 

45 

16 

800 

472 

.0259 

610 

44 

17 

821 

529 

.0253 

590 

43 

18 

842 

586 

.0247 

569 

42 

19 

862 

643 

.0241 

549 

41 

20 

. 69883 

. 97700 

1 . 0235 

.71529 

40 

21 

904 

756 

.0230 

508 

39 

22 

925 

813 

.0224 

488 

38 

23 

946 

870 

.0218 

468 

37 

24 

966 

927 

.0212 

447 

36 

25 

. 69987 

.97984 

1 . 0206 

.71427 

35 

26 

.70008 

.98041 

.0200 

407 

34 

27 

029 

098 

.0194 

386 

33 

28 

049 

155 

.0188 

366 

32 

29 

070 

213 

.0182 

345 

31 

30 

. 70091 

. 98270 

1.0176 

71325 

30 

31 

112 

327 

.0170    305 

29 

32 

132 

384 

.0164    284 

28 

33 

153 

441 

.0158    264 

27 

34 

174 

499 

.0152    243 

26 

35 

.70195 

. 98556 

1.0147  .71223 

25 

36 

215 

613 

.0141 

203 

24 

37 

236 

671 

.0135 

182 

23 

38 

257 

728 

.0129 

162 

22 

39 

277 

786 

.0123 

141 

21 

to 

. 70298 

. 98843 

1.0117 

.71121 

20 

41 

319 

90  i 

.0111    100 

19 

42 

339 

. 98958 

.0105    080 

18 

43 

360 

.99016 

. 0099    059 

17 

44 

381 

073 

.00u4    039 

16 

4i 

.70401 

.99131 

1.0088  .71019 

15 

46 

422 

189 

.0082  .70998 

14 

47 

443 

247 

.0076    978 

13 

48 

463 

304 

.0070    957 

12 

49 

484 

362 

.0064    937 

11 

50 

. 70505 

. 99420 

1.0058  .70916 

IO 

51 

525 

478 

.00521   896 

9 

52 

546 

536 

.0047:   875 

8 

53 

567 

594 

.0041    855 

7 

54 

587 

652 

.0035,   834 

6 

55 

. 70608 

.99710 

1.0029^.70813 

5 

56 

628 

768 

.  002:^    793 

4 

57 

649 

826 

.0017    772 

3 

58 

670 

884 

.0012    752 

2 

59 

690 

. 99942 

.0006    731 

1 

60 

.70711 

1^.0000  1.0000  .70711 

O 

NCOS 

N  Cot:  N  Tar 

l  NSin 

/ 

45° — Natural  Functions 


TABLE  IV 

Table  of  Powers  and  Roots 


V* 


v 


^ 


ivi 


Table  of  Powers  and  Roots 


101 


No. 

Squares 

Cube,    |Xt? 

Cube 
Roots 

No. 

Squares 

Cubes 

Square 
Roots 

Cube 
Roots 

1 
2 
3 
4 

1 

4 

,9 

16 

1 

8 
27 
64 

1.000 
1.414 
1.732 
2.000 

1.000 
1.259 
1.442 
1.587 

51 
52 
53 
54 

2,601 
2,704 
2,809 
2,916 

132,651 
140,608 

148,877 
157,464 

7.141 
7.211 
7.280 
7.348 

3 .  708 
3.732 
3.756 
3.779 

5 

6 

7 
8 
9 

25 
36 
49 
64 
81 

125 
216 
343 
512 
729 

2.236 
2.449 
2.645 

2.828 
3.000 

1.709 
1.817 
1.912 
2.000 
2.080 

55 

56 

57 

58 

-   59 

3,025 
3,136 
3,249 
3,364 
3,481 

166,375 
175,616 
185,193 
195,112 
205,379 

7.416 
7.483 
7.549 
7.615 
7.681 

3.802 
3.825 
3.848 
3.870 
3.892 

io 

11 
12 
13 
14 

100 
121 
144 
169 
196 

1,000 
1,331 
1,728 
2,197 
2,744 

3.162 
3.316 
3.464 
3.605 
3.741 

2.154 
2.223 
2.289 
2.351 
2.410 

60 

61 
62 
63 
64 

3,600 
3,721 
3,844 
3,969 
4,096 

216,000 
226,981 
238,328 
250,047 
262,144 

7.745 
7~M0 
7.874 
7.937 
8.000 

3.914 
3.936 
3.957 
3.979 
4.000 

15 

16 

17  - 

18 

19 

225 
256  . 
289 
324 
361 

3,375 
4,096 
4,913 
5,832 
6,859 

3.872 
4.000 
4.123 
4.242 
4.358 

2.466 
2.519 
2.571 
2.620 
2.668 

65 

66 
67 

68 
69 

4,225 
4,356 

4,489 
4,624 
4,761 

274,625 
287,496 
300,763 
314,432 
328,509 

8.062 
8.124 
8.185 
8.246 
8.306 

4.020 
4.041 
4.061 
4.081 
4.101 

20 

21 
22 
23 
24 

400 
441 
484 
529 
576 

8,000 

9,261 

10,648 

12,167 

13,824 

4.472 
4.582 
4.690 
4.795 

4.898 

2.714 

2.758 
2.S02 
2.843 

2.884 

TO 

71 
72 
73    . 

74 

4,900 
5,041 
5,184 
5,329 
5,476 

343,000 
357,911 
373,248 
389,017 
405,224 

8.366 
8.426 
8.485 
8.544 
8.602 

4.121 
4.140 
4.160 
4.179 
4: 198 

25 

26 
27 
28 
29— 

625 
676 
729 

784 
841 

15,625 
17,576 
19,683 
21,952 
24,389 

5.000 
5.099 
5.196 
5.291 
5.385 

2.924' 
2.962 
3.000 
3.036 
3.072 

75 

76 

77 
78 
79 

5,625 
5,776 
5,929 
6,084 
6,241 

421,875 
438,976 
456,533 
474,552 
493,039 

8.660 
8.717 
8.774 
8.831 
8.888 

4.217 
4.235 
4.254 
4.272 
4.290 

30 

31 
32 
33 
34 

900 

961 

1,024 

1,089 

1,156 

27,000 
29,791 
32,768 
35,937 
39,304 

5.477 
5.567 
5.656 
5.744 
5.830 

3.107 
3.141 
3.174 
3.207 
3.239 

SO 
81 
82 
83 

84 

6,400 
6,561 
6,724 
6,889 
7,056 

512,000 
531,441 
551,368 
571,787 
592,704 

8.944 
9.000 
9.055 
9.110 
9.165 

4.308 
4.326 
4.344 
4.362 
4.379 

35 

36 
37 
38 
39 

1,225 
1,296 
1,369 
1,444 
1,521 

42,875 
46,656 
50,653 
54,872 
59,319 

6.916 
6.000 
6.082 
6.164 
6.244 

3.271 
3.301 
3.332 
3.361 
3.391 

85 
86 
87 
88 
89 

7,225 
7,396 
7,569 

7,744 
7,921 

614,125 
636,056 
658.503 
681,472 
704,969 

9.219 
9.273 
9.327 
9.380 
9.433 

4.396 
4.414 
4.431 
4.447 
4.464 

40 

41 

42 
43 

44 

1,600 
1,681 
1,764 
1,849 
1,936 

64,000 
68.921 
74,088 
79,507 

85,184 

6.324 
6.403 
6.480 
'6 .  557 
6.633 

3.419 
3.448 
3.476  | 
3.503 
3.530 

90 

91 
92 
93 
94 

8,100 
8,281 
8,464 
8,649 
8,836 

729,000 
753,571 

778,688 
804,357 
830,584 

9.486 
9.539 
9.591 
9.643 
9.695 

4.481 
4.497 
4.514 
4.530 
4.546 

45 

46 
47 
48 
49 

2,025 
2,116 
2,209 
2,304 
2,401 

91,125 

97,336 

103,823 

110,592 

117,649 

6.708 
6.782 
6.855 
6,928 
IT  9.28 

3.556 
3.583 
3.608  I 
3.634  | 
3.659 

95 

96 
97 
98 
99 

9,025 

9,216 
9,409 
9,604 
9,801 

857,375 
884,736 
912.673 
941,192 
970,299 

9.746 
9.797 
9.848 
9.899 
9.949 

4.562 
4.578 
4.594 
4.610 
4.626 

50 

2,500 

125,000 

7.071 

3.684 

lOO 

10,000 

1,000,000 

1Q„  000 

4.641 

TABLE  V 

Formulas 


Formulas  105 

PLANE   GEOMETRY 

1.  Length  of  circle  =  2irr  =  3. 141 59d 

2.  Area  of  circle     =irr2 

3.  Area  of  triangle  =  \bh  =  \ab  sin  C  =  \r(a-\-b+c) 
abc 


=  ^  =  V/s(s-a)(s-b)(s-c) 


4.  Area  of  parallelogram  =  bh 

5.  Area  of  square  =  a2 

6.  Area  of  equilateral  triangle  =  —  V3 

7.  Area  of  trapezoid  =  J/i(&i+&2)  =  hm 


SOLID   GEOMETRY 

1 .  Volume  of  prism  =  ba 

2.  Volume  of  pyramid  =  ^ba 

3.  Volume  of  right  circular  cylinder  =  irr2(\ 

4.  Total  surface  of  right  circular  cylinder  =  2irr (r+  A) 

5.  Lateral  surface  of  right  circular  cylinder  =  2irrd 

6.  Volume  of  right  circular  cone  =  §7rr2A 

7.  Lateral  surface  of  right  circular  cone  =  7rrs 

8.  Total  surface  of  right  circular  cone  =  7rr(r+s) 

9.  Surface  of  sphere  =  47rr2 

4 
10.  Volume  of  sphere  =  ^jrr3 

series 

1.  Arithmetical  progression : 

ft 
l  =  a-\-(n—l)d;    s  =  ~(a+l) 


106  Formulas 


2.  Geometrical  progression : 

,        „   ,  a  —  arn       .„       .        .  a 

l  =  arn~x\     s=— :     if  r<  1  and  n->oo  ,  s  =  - 

1-r  1-r 

3.  Binomial  theorem : 

{a+b)n  =  an+n.an-'b+n{\~})an-W 
1  1  •  A 

+n(n-^2)an.t6,+ete 

The  ftth  term 

n(n-l)(n-2)  ....  (n-/c+2)__,4,1^_1 
= L2.3.  ..  .k-1 a       +b 


LOGARITHMS 

1.  log  ab  =  log  a+log  b 
p.  log  r  =  log  a  —  log  b 

3.  log  an  =  n  log  a 

.    .       */-    log  a 

4.  log  V  a  =  —^~ 

n 

5.  log  1  =  0 
logfciV 


6.  loga  N  = 


log6  a 


7.  cologiV  =  log^=(10-logiV)-10 


QUADRATIC   EQUATION 

If  az2+fcc+c  =  0,  a;  =  — ^ . 

If  62  —  4ac  =  0,  the  roots  are  real  and  equal. 
If  b2  —  4ac>0,  the  roots  are  real  and  unequal. 
If  b2  —  4ac<0,  the  roots  are  complex. 


Formulas  107 


TRIGONOMETRIC    FORMULAS 


aba 
1.  sin  a  =  -,  cos  a  =  - ,  tan  a  =  r, 
c  c  b 

c  c  b 

esc  a  =  - ,  sec  a  =  t  ,  cot  a  =  - 

a1  6  a 


2.  sin2  a  +  cos2  a  =  l  a  cos  a 

6.  cot  a  =  - 

3.  sec2  a  =  l+tan2  a 

1 

4.  csc2a=l  +  cot2a  '•  sec  a  =  ^oTa 

sin  a  0 

5.  tan  a  = 8.  esc  a 


cos  a  sin  a 


9.  sin  (a  =*=£)=  sin  a  cos  /3=*=cos  a  sin  (3 
10.  cos  (a  =*=  j8)  =  cos  a  cos  /3  =f  sin  a  sin  (3 
tan  a^tan  /?. 


11.  tan(a±0)  = 


1  =f  tan  a  tan  /3 


12.  sin  a+sin  (3  =  2  sin  J(a+j8)  cos  |(a  — /3) 

13.  sin  a  —  sin  |S  =  2  cos  |(a-f /?)  sin  ^(a  — /3) 

14.  cos  a+cos  0  =  2  cos  J(a+^)  cos  2(a  — 0) 

15.  cos  a  —  cos  0  =  —  2  sin  J(a+/3)  sin  -§(a  —  jS) 

16.  sin  a  sin  /3  =  ^  cos  (a  —  ff)—  \  cos  (a+/5) 

17.  cos  a  cos  0  =  \  cos  (a  —  j8)+J  cos  (a+/3) 

18.  sin  a  cos  /?  =  J  sin  (a+/3)+J  sin  (a  —  /3) 

19.  sin  2  a  =  2  sin  a  cos  a 

20.  cos  2  a  =  cos2  a  —  sin2  a 

=  2cos2a-l  =  l-2sin2a 


108  Formulas 

rti  0         2  tan  a  29.  cos  (7r±0)  =  —  cos  0 

21.  tan  2  a  =  - — - — r 

1-tana  3()    tan  (tt  =«=  ^)  =  =«=  tan  ^ 

22.  sin±a=^  JE|*1*  3L  sin  (r**)— ■** 

32.  cos  ( —  x)  =  cos  z 

23.  cos^a^JH^  33-  tan  (-*)  = -tans 

34.  esc  (  —  x)  =  —  esc  x 

24.  tanta=^A/1~cosa        '  35.  sec  (-x)=secx 

2  \l+cosa  on  ,       N 

36.  cot  ( —  x)  =  —  cot  x 

25.  sin  f  |  ±  0 J  =  cos  (9  37.  sin  30°  =  J 

v  38.  sin  45°  =  §1/2 

26.  cos(j^0j  -  =f  sin  0  39>  gin  600  =  1^3 

/x      \  40.  cos  30°  =  \V  3 

27.  tan (-=±=0    =  ^cot0  ^         ■     0     -  /- 

\2      /  41.  cos  45°  =  §1/2 

28.  sin  (tt±0)  =  =Fsin  0  42.  cos  60°  =  \ 

TRIANGLES 

AO       a  b  c  Aa    a-b    sin§(A-£) 

4o.    -: -r  =  — n  =  — ~Fi  40.    = j— ^ 

sin  A     sin  B     sin  C  c  cos  f  C 

44.  a2  =  62+c2-26c  cos  A  a+6  =  cos  §  (A-B) 

a+6  =  tan§  (A+B)  "      c  sin  \  C 

a  —  b    tan  §  (A  —  B) 

Us  =  i  (a+b+cy.  y 

48.  sin  \  A  =  ^is~h)b(cS~c)  ■    49.  cos  \  A=^8~a) . 

50.  tan  jA-Jb-M'-Zc} 
2        \     s(s-a) 


Formulas  109 

If  r  =  radius  of  inscribed  circle : 

\  s  2         s-a 

53.  tani£  =  -^.  54.  tan£C  =  ~. 

__     .  ,     ,    .     ~     c2     sin  A  sin  5 

55.  Area  =  §  ao  sin  C  =77  •  : — 7= — 

2  sin  C 

=  l/s(s-a)(s-6)(s-c) 

56.  Diameter  of  circumscribed  circle 

a  b  c 

sin  A     sin  £     sin  C 


TABLE  VI 

Equivalents  and  Logarithms  of  Important 
Constants 


Equivalents  and  Logarithms  of  Important  Constants    113 


TCnmhpr                       Common 
.Number                    Logarithm 

3.14  159  6 
57.29  577  9 

0.01  745  3 
2.71  828  2 
0.43  429  4 
2.30  258  5 

1 .  60  934  7  kilom. 
0.91  440  2  meter 
0.30  480  1  meter 

25 .  40  005  mm. 

39 .  37  inches 
1.09  361  1  yard 
3 .  28  083  3  feet 
6080. 290  feet 

0.62  137  Omile 

7000  grains 
453 .  59  242  8  grammes 
28 .  34  953  grammes 
31 .  10  348  grammes 
0 .  06  479  9  gramme 
2.20  462  2  lbs.  Av. 
15.43  235  6  grains 

1.05  668  U.S.  quart 
0.26  417  U.S.  gal. 

33.814  U.S.  fluid  oz. 
0.94  636  liter 
3.78  544  liters 
0.02  957  3  liter 
231  cu.  inches 
4.54  346  liters 

36.34  77  liters 

0.49  715  0 
1.75  812  2 

2.24  187  7 
0.43  429  4 
1.63  778  4 
0.36  221   6 

0.20  665  0 

1.96  113  7 

1.48  401   6 
1.40  483  5 
1.59  516  5 
0.03  886  3 
0.51  598  4 

3.78  392  4 

1.79  335  0 

3.84  509  8 
2.65  666  6 
1.45  254  6 

1.49  280  9 
2.81   156  8 
0.34  333  4 
1.18  843  2 

0.02  394  4 
1.42  188  4 
1.52  910 

1.97  605  6 
0.57  811   6 
2.47  090 
2.36  361   2 
0.65  738  7 
1.56  047  7 

1  radian  =i^ 

IT 

1  degree  =z-^.  radians 

loO 

to  (modulus  of  common  loga- 

TO 

1  mile 

1  yard 

1  foot 

1  inch 

1  nautical  mile 

1  kilometer 

1  pound  Av 

1  ounce  Av 

1  ounce  Troy 

1  grain 

1  kilogramme 

1  gramme 

1  liter 

1  quart,  U.S 

1  gallon.  U.S 

1  fluid  ounce.  .  . . 

1  gallon  U.S 

1  British  gallon 

1  British  bushel 

TABLE  VII 

Reductions 


Degrees,  Minutes,  and  Seconds  Reduced  to  Radians    117 


0 

Radians 

' 

Radians 

" 

Radians 

1 

2 
3 
4 

0.01  745  33 
0.03  490  66 
0.05  235  99 
0.06  981  32 

1 
2 
3 

4 

0.00  029  09 
0.00  058  18 
0.00  087  27 
0.00  116  36 

1 
2 
3 
4 

0.00  000  48 
0.00  000  97 
0.00  001  45 
0.00  001  94 

5 

6 

7 
8 
9 

0.08  726  65 
0.10  471  98 
0.12  217  30 
0.13  962  63 
0.15  707  96 

5 

6 

7 
8 
9 

0.00  145  44 
0.00  174  53 
0.00  203  62 
0.00  232  71 
0.00  261  80 

5 

6 

7 
8 
9 

0.00  002  42 
0.00  002  91 
0.00  003  39 
0.00  003  88 
0.00  004  36 

io 

0.17  453  29 

IO 

0.00  290  89 

IO 

0.00  004  85 

20 

0.34  906  59 

15 

0.00  436  33 

15 

0.00  007  27 

30 

0.52  359  88 

20 

0.00  581  78 

20 

0.00  009  70 

40 

0.69  813  17 

25 

0.00  727  22 

25 

0.00  012  12 

50 

0.87  266  46 

30 

0.00  872  66 

30 

0.00  014  54 

60 

1.04  719  76 

35 

0.01  018  11 

35 

0.00  016  97 

70 

1.22  173  05 

40 

0.01  163  55 

40 

0.00  019  39 

80 

1.39  626  34 

50 

0.01  454  44 

50 

0.00  024  24 

90 

1.57  079  63 

60 

0.01  745  33 

60 

0.00  029  09 

Reduction  of  Minutes  to  Degrees 


0'  = 

=  0?000  !lO'  = 

=0?166 

20  = 

=0?333 

30'  = 

=0?500 

40  = 

=0?666i5O'  = 

=0?833 

V 

.016  11' 

.183 

21' 

.350 

31' 

.516 

41' 

.683  !  51' 

.850 

2' 

.033  12' 

.200 

22' 

.366 

32' 

.533 

42' 

.700 

52' 

.866 

3' 

.050  13' 

.216 

23' 

.383 

33' 

.550 

43' 

.716 

53' 

.883 

4' 

.066  14' 

.233 

24' 

.400 

34' 

.566 

44' 

.733 

54' 

.900 

5' 

.083  15' 

.250 

25' 

.416 

35' 

.583 

45' 

.750 

55' 

.916 

6' 

.100  16' 

.266  26' 

.433 

36' 

.600!  46' 

.766  56' 

.933 

7' 

.116  17' 

.283  27' 

.450 

37' 

.616  47' 

.783  57' 

.950 

8' 

.133  18' 

.300 

28' 

.466  1  38' 

.633 

48' 

.800 

58' 

.966 

9' 

.150 

19' 

.316 

29' 
30'  = 

.483 
=0?500 

39' 

.650 

49' 

.816 

59' 
60'  = 

.983 
=1?000 

Reduction  of  Seconds  to  Degrees 


6"  =0?00166 
7"  =0.00194 
8"  =000222 
9"  =0  ?0025C 


10"  =0?00277 
15"  =0.00416 
20"  =0.00555 
30"  =0?00833 


35"  =0!00970 
40"  =0.01111 
45"  =0.01250 
50"  =0?01388 


118     Reduction  of  Degrees  to  Minutes  and  Seconds 


o?oo 

=   0' 

0?3©=18' 

0?60  =36' 

0?90  =54' 

.01 

0'36" 

.31       18'36" 

.61       36'36" 

.91       54'36" 

.02 

1/12" 

.32       19'12" 

.62      37'12" 

.92       55'12" 

.03 

1'48" 

.33       19'48" 

.  63      37'48" 

.93       55'48" 

.04 

2'24" 

.  34       20'24" 

.64      38'24" 

.  94       56'24" 

0?05 

3' 

0?35  =21' 

0?65  =39' 

0?95  =57' 

.06 

3'36" 

.36       21'36" 

.66       39'36" 

.96      57'36" 

.07" 

4'12" 

.37       22'12" 

.67       40'12" 

.97       58'12" 

.08* 

4'48" 

.38       22'48" 

.  68      40'48" 

.  98      58'48" 

.09 

5'24" 

.39      23'24" 

.69      41'24" 

.  99      59'24" 

o?io 

-   6' 

0?40  =24' 

0?70  =42' 

1?00  =60' 

.11 

6'36" 

.41       24'36" 

.71       42'36" 

.12 

7'12" 

.42       25'12" 

.72       43'12" 

.13 

7'48" 

.43       25'48" 

.73      43'48" 

.14 

8'24" 

.44      26'24" 

.  74      44'24" 

0?15 

-   9' 

0?45  =27' 

0?75  =45' 

.16 

9'36" 

.  46       27'36" 

.76      45'36" 

o?ooo  =  or 

.17  • 

10'12" 

.47       28'12" 

.77       46'12" 

.001          3  76 

.18 

10'48" 

.48       28'48" 

.  78      46'48" 

.002          7f2 

.19 

11 '24" 

.49       29'24" 

.79      47'24" 

.003        10*8 
.004        14  *  4 

0?20 

=  12' 

0?50  =30' 

0?80  =48' 

.21 

12'36" 

.51       30'36" 

.81       48'36" 

0?005  =18f 

.22 

13'12" 

.52      31'12" 

.82      49'12" 

.006       21  f6 

.23 

13'48" 

.53      31'48" 

.  83       49'48" 

.007        25  ".2 

.24 

14'24" 

.54      32'24" 

.  84       50'24" 

.008        28  f  8 
.009       32  f4 

O?2o 

=  15' 

0?55  =33' 

0?85  =51' 

.26 

15'36" 

.56"     33'36" 

.86      51'36" 

0?01      =36f 

.27 

16'12" 

.57       34'12" 

.87       52'12" 

.28 

16'48" 

.  58      34'48" 

.88      52'48" 

.29 

17'24" 

.59      35'24" 

.89      53'24" 

0?30 

=  18' 

0?60  =36' 

0?90  =54' 

INDEX 


INDEX 


[References  are  to  sections,  not  to  pages] 


Abridged  division 139 

Abridged  multiplication .  .  .  133 

Addition  theorem 195 

Angle:  between  two  curves  332 

cosine  of 32 

Angles:  in  general 29 

polyedral 320 

sine  of 32 

spherical 332 

tangent  of 32 

triedral 321 

Area :  of  a  cone 272 

of  a  cylinder 272 

of  a  frustum  of  a  cone. . .  273 

of  a  prism 269 

of  a  pyramid 270 

of  a  sphere 280 

of  oblique  triangle 192 

Arithmetical  means 214 

Arithmetical  progression. .  .  211 

Axis  of  a  cone 250 

Bernoulli,  James 5 

Bernoulli,  John 5 

Bezout 72 

Binomial  theorem 206 

Cauchy 72 

Cavalieri's  theorem 302 

Characteristic 147 

Circle,  equation  of 226 

Cologarithm 188 

Common  logarithms 146 

Complex  fractions . 104 


Complex  numbers 89 

Cone:  altitude  of 248 

axis  of 250 

base  of 248 

circular 250 

frustum  of 267 

lateral  surface  of 272 

of  revolution 250 

right  circular 250 

sections  of 268 

similar.  .  . 275 

Conical  surface.  .• 248 

directrix  of 248 

elements  of 248 

generatrix  of 248 

vertex  of 248 

Constant 2 

Cosines,  law  of 181 

Cramer 72 

Cube.. 263 

Cylinder:  altitude  of 251 

base  of 251 

circular 251 

elements  of 251 

lateral  area  of 269 

of  revolution 252 

similar 274 

volume  of 298,  299 

Denominator,  rationalizing.  129 

Determinant 75 

solution  by 76 

Diameter  of  circumscribed 

circle 180 


121 


122 


THIRD-YEAR  MATHEMATICS 


[References  are  to  sections,  not  to  pages] 


Direct  variation 11 

Directrix:   of  a  conical  sur- 
face. .  . , 248 

of  a  cylindrical  surface. . .  251 

Discriminant 92 

Division :  abridged 139 

synthetic 16 

Dodecaedron 245 

Elements  of  a  progression. .  212 

Ellipse,  equation  of 228 

Equations:  equivalent 73 

exponential 162 

formation  of 93 

homogeneous 236 

inconsistent 73 

irrational 132,238 

linear 67,68,70 

of  quadratic  form 87 

quadratic,     in     one    un- 
known  81,  83 

solution  by  factoring ....     22 

solution  by  graph 71 

trigonometric. ...  88,  134,  202 

with  imaginary  roots. ...     89 

Evaluation  of  functions.  . .  6,  21 

Excess,  spherical 340 

Exponents:  laws  of 117 

fractional 116 

negative 115 

zero 114 

Exponential  equations 162 

Factor  theorem 22 

Factorial  notation 208 

Factoring 94-102 

Fractional  equations 238 


Fractions,  complex 104 

Frustum:  of  a  cone 267 

of  a  pyramid 265 

slant  height  of 266 

Function 1,4 

cubic 14 

evaluation  of 6 

graph  of  linear 8 

inverse 37 

linear 7,  9 

line — representation  of .  .     39 


of  f--.) 


60 


62 


of  double  an  angle 199 

of  half  an  angle 200 

of  negative  angles 57 

quadratic 12, 13 

trigonometric 32 

Functional  notation 5 

Gauss 72 

General      quadratic      equa- 
tion   225 

Generatrix,  of  a  conical  sur- 
face   248 

of  a  cylindrical  surface. . .  251 

Geometrical  means 218 

Geometrical  progression ...  215 

elements  of 216 

infinite 220 

Graph:  of  cubic  function.  .     14 

of  linear  function 8 

of  logarithmic  function .  .    148 
of  quadratic  function. ...     13 
of     trigonometric     func- 
tions       50 


INDEX 


123 


[References  are  to  sections,  not  to  pages] 


Graphical    solution:     of    a 

linear  system 71 

of  a  quadratic  system  in 
two  variables 233 

Hyperbola 229 

Icosaedron 245 

Imaginary  numbers 90 

Infinite  geometrical  series . .  220 

Intercepts 10 

Inverse  functions 37 

Inverse  variation 24 

Irrational  equations. .  .    132,  238 

Irrational  numbers 90 

Joint  variation 26 

Lagrange 72 

Laplace 72 

Law  of  cosines 181 

of  sines 179 

of  tangents 183 

Laws  of  exponents 117 

Leibnitz 5,  72 

L'Hopital 72 

Linear  equations  in  one  un- 
known    67,  68 

Linear  function 7,  9 

Line  representation  of  func- 
tions       39 

Logarithms 143 

common 146 

of  a  power 158 

of  a  product 156 

of  a  quotient 157 

of  a  root 159 

table  of 152, 170 

Lune 351 


Mantissa 147 

Means:    arithmetical 214 

geometrical .  218 

Measure,  radian 46 

Measurement,  precision  of .  137 

Mollweide's  equations  ....  184 

Multiplication,  abridged . . .  138 

Napier 145 

Notation,  functional 5 

Oblique  triangle:   area  of..   192 

solution  of 187 

Octaedron 245 

Parabola 13 

Parallelopiped:  oblique.  .  .    263 

rectangular. 263 

right 263 

volume  of 286,  291 

Polar  triangles 336 

Polyedral  angle 320 

Polyedron 243 

Polyedrons 245 

regular 325 

Polygon,  spherical 323 

Precision  of  measurement . .  137 

Prism :  altitude  of 253 

bases  of 253 

lateral  area  of 269 

lateral  edges 253 

lateral  faces  of 253 

oblique 254 

right 254 

truncated 264 

volume  of 294 

Progression:  arithmetical..  211 
geometrical 215 


124 


THIRD- YEAR  MATHEMATICS 


[References  are  to  sections,  not  to  pages] 


Pyramid 247 

altitude  of 247 

base  of 247 

frustum  of 265 

inscribed  in  a  cone 295 

lateral  area  of 270 

lateral  edges  of 247 

lateral  faces  of 247 

regular 249 

vertex 247 

volume  of 304 

Quadrant 31 

Quadratic  equation 81 

graph  of  quadratic  in  two 

unknowns 227 

in  two  unknowns 225 

nature  of  the  roots  of . . .  .     92 

Quadratic  function 12,  13 

simultaneous 232 

solution    by    completing 

square 83 

solution  by  formula 83 

Radian  measure 46 

Radical 120 

reduction  of 121 

Rationalizing  denominator .   129 

Regular  polyedrons 325 

Relation  between  the  roots 

and  coefficients 93 

Remainder  theorem 19,  20 

Right  section 256 

Roots  of  a  quadratic  equa-     89 
tion    relation    between 
roots  and  coefficients .  .     93 

square  roots 84 

square  root  of  a  radical 
expression 131 


Section 255 

of  a  cone 268 

Segment,  spherical 310 

Signs  of  functions 33 

Similar  cones 275 

Similar  cylinders 274 

Simultaneous  quadratics . . .  232 

Sines,  law  of 180 

Slant  height,  of  a  frustum  of 

a  pyramid 266 

Slide  rule 163 

Solution:     of    oblique    tri- 
angles    187 

of  right  triangle 174 

Sphere :  surface  of 280 

volume  of 309 

Spherical  cone 313 

Spherical  excess 316 

Spherical  polygon 323 

Spherical  sector 315 

Spherical  segment 310 

Spherical  triangle 335 

Square  root :  of  polynomials  84 

of  a  radical  expression.  . .  131 

Subtraction  theorem 195 

Sums  of  sines  and  cosines.  197 
Symmetrical  polyedral 

angles 342 

Symmetrical  polygons 343 

Synthetic  division 16 

Table  of  logarithms .  .  .    152,  170 

Tangent :   of  angle 32 

of  half -angle 185 

law  of 183 

Tetraedron 245 

Triangle:    birectangular.  .  .  335 

polar 236 


INDEX 


125 


[References  are  to  sections,  not  to  pages] 


spherical 335 

solution  of  right 174 

trirectangular 335 

Triedral  angle 321 

Trigonometric  equations 

88,  134,  202 

Trigonometric  functions ...  32 

Truncated  prism 264 

Variable 3 

Variation:  direct 11 


direct  and  inverse 27 

inverse 24 

joint 2G 

Volume :  of  a  cone 306 

of  a  cylinder 298 

of  a  prism 294 

of  a  pyramid 304 

of  a  sphere 309 

of  frustum  of  a  cone ....   307 


Zone. 


281 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

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REC'D  LD 


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MAR  2  3  1968    $ 


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